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COPYRIGHT DEPOSIT. 



JUNIOR HIGH SCHOOL SERIES 

INTRODUCTION TO 
MATHEMATICS 

BY 
ROBERT L. SHORT 

PRINCIPAL OF WEST TECHNICAL HIGH SCHOOL 
CLEVELAND, OHIO 

AND 

WILLIAM H. ELSON 

FORMER SUPERINTENDENT OF SCHOOLS 
CLEVELAND, OHIO 



D. C. HEATH & CO., PUBLISHERS 
BOSTON NEW YORK CHICAGO 






Copyright, 1916, 
By D. C. Heath & Co. 

ih6 



AUG 29 1916 



CIA438176 



INTRODUCTION 

This book is intended as an aid in the movement to vitalize 
mathematics, adapting it to the needs and the understanding 
of pupils. It employs the increasingly popular correlated 
method, combining related portions of arithmetic, algebra, and 
geometry. It treats these branches of mathematics more with 
reference to their unities and less as isolated entities (sciences). 
It seeks to give pupils usable knowledge of the principles 
underlying mathematics and ready control of them. 
. The combined arrangement not only increases interest and 
motivates the work, but it also gives greater power of analysis 
on the part of the learner and greater accuracy in results. 
The early study of geometry brings analysis into play at every 
step and stage ; consequently written problems to be stated 
have no terrors for those who are taught in this way. 

Growing discontent with mathematics as traditionally taught, 
in view of the large number of failing pupils, has led mathe- 
matical associations to urge teachers to select and stress those 
portions of mathematics that are vitally useful. For example, 
these associations have for several years recommended that all 
work should be based upon the equation. In accordance with 
this view we have made the demonstrations in this book largely 
algebraic, thus making the demonstration essentially a study 
in simultaneous equations. 

In this method of treatment, we have found it advantageous 
not to hurry the work. Pupils gain power, not in solving a 
great many problems, but in analyzing and thoroughly under- 
standing the principles of a few. 

In general, the book covers straight line geometry to propor- 
tion and algebra through fractional equations j it is intended 
for one year's work. 

iii 



iv INTRODUCTION 

We are indebted to many who have offered suggestions and 
practical problems, and especially to Carlotta Greer, of East 
Technical High School, Cleveland, Ohio, Professor Kenneth G. 
Smith, of the Iowa State College, John W. Thalman, B. C. 
Smith, and H. E. Garner, of West Technical High School, 
Cleveland, Ohio, and also to those who so kindly read the 
proof sheets. 



TABLE OF CONTENTS 



BOOK I 

CHAPTER PAGE 

I. The Number System 1 

Addition 3 

Prime Factors 5 

Oral Control of Number 8 

II. Equations 10 

Problems 13 

III. Positive and Negative Numbers . . .17 

The Four Fundamental Operations . . . 18-29 

Supplemental Applied Mathematics .... 29 

IV. Polynomials, Multiplication 36 

Polynomials, Division 41 

Review . . .46 

Supplemental Applied Mathematics .... 48 

V. Inequalities . . 54 

Simultaneous Equations .... « o 56 

Problems 64 

Supplemental Applied Mathematics 66 

VI. Lines, Angles, Triangles 69 

Supplemental Applied Mathematics .... 99 

VII. Graphs, the Algebra of Lines. Parallels and 

their Uses . 103 

Quadrilaterals 120 

Polygons 127 

Supplemental Applied Mathematics . . . .130 

v 



VI TABLE OF CONTENTS 

CHAPTER PAGE 

VIII. Products and Factors 133 

(a + b) (a - h) 133 

(a + by 135 

X 2 + fa; + C 137 

Common Factors 138 

ax 2 + bx + c 140 

a 3 ±& 3 . 142 

x* + bx 2 + c 2 . . .144 

Solutions by Factoring 148 

Supplemental Applied Mathematics . . . .151 

IX. Fractions 155 

Multiples 159 

Addition 160 

Multiplication 166 

Division 168 

Equations 170 

Supplemental Applied Mathematics .... 181 

X. Proportion . . . 188 

List of Constructions 193 

List of Theorems 194 

Index 198 



FOR THE TEACHER 

Keviews in mathematics are always necessary. This is especially 
true in this text, which combines different branches of mathematics. 

In teaching the text keep in mind that in geometry as well as in 
algebra problems are solved by means of equations. The equation is 
the principal tool used. To use the equation method successfully, the 
Hilbert notation, a small letter for angle values and for line values is 
essential. 

In lettering a figure, begin at the lower left-hand corner and read 
counter-clockwise. This gives pupils an idea of directed lines, and 
makes possible the correct drawing of the figure from description. 

Theorems I and II should be assumed as true until after Theo- 
rem IX. If they are then proved, the student will not be so apt to 
attempt to prove every theorem and problem by superposition. 

Note that many demonstrations have been put into the form of a 
set of simultaneous equations, the solution of which produces the 
desired equation. 

Emphasize the manipulation of quantities by means of factors and 
the use of methods of indication until there is no longer hope that 
the factors may disappear through division (see p. 178). 

Do a large amount of the work orally, and do it so often that the 
pupil knows what he is doing, and why he is doing it. No pupil 
should use pencil and paper to find prime factors of (24) 2 , (12) 6 , 9 ■ 27, 
or to find the product of 18 • 17. 

Insist that pupils study all illustrative work, rules, and instructions 
before beginning the examples of an exercise. 

Teach pupils to use the Index, also the groups of theorems and 
constructions found on pages 193-197. 



vu 



INTRODUCTION TO 
MATHEMATICS 



CHAPTER I 

The Number System 

1. Our number system is a decimal one. Ten units of one 
order make one of the next higher. One tenth of any digit 
makes a digit of the next lower order. The digits are 1, 2, 
3, 4, 5, 6, 7, 8, 9. All numbers are made up of these digits 
and their position is often indicated by the introduction of one 
other symbol, 0, known as nought, cipher, or zero; e.g., 16 is 
equal to ten l's added to six l's. That is, 16 = 10 + 6, the 
position of the 1 being indicated by the 0. 

The numbers expressed by these digits themselves are often 
the multiples of other digits. For example, 4 = two 2's, or 
2 • 2, where the • indicates multiplication. 

6 = 3-2, 9 = 3-3. 

The numbers 1, 2, 3, 5, 7 are the prime digits. A prime num- 
ber is a number whose only integral factors are itself and unity. 

EXERCISE l 

Write the following numbers in such a way that their deci- 
mal composition will appear : 



1. 145. 




8. 511. 


145 = 100 + 40 + 5- 






2. 223. 4. 987. 


6. 227. 


9. 101. 


3. 448. 5. 999. 


7. 863. 


10. 10016. 



2 MATHEMATICS 

2. In the product of two or more numbers, any one of them 
or the product of any number of them is a factor of the given 
product 2 • 3 • 5 = 30. Then, 2 is a factor of 30. 2 . 3 or 6 
is also a factor of 30.* 

3. A term is a number whose parts are not separated by the 
plus (+) or minus (— ) sign. 

In the expression, 

10 + 6 
10 is a term. 6 is also a term. 
10 + 6 is composed of two terms. 

4. A binomial is an expression of two terms. 
A trinomial is an expression of three terms. 

A quadrinomial is an expression of four terms. 

An expression of two or more terms is also called a poly- 
nomial. 

100 + 60 + 3 is a trinomial. 

5. It is often necessary to represent a number by a letter or 
a combination of letters. Such letters may represent either 
unknown numbers or those supposed to be knoivn numbers. 
This kind of notation is used in general arithmetic or algebra. 

E.g., n may represent any number, likewise any letter or 
combination of letters and figures may be considered a number. 

a + b + c is a trinomial number (§ 4), or the sum of three 
numbers a, b, and c. In arithmetic it is possible to express 
such a sum as a single number. 

Thus, 2 + 5 + 8 = 15. 

In algebra, this is not possible unless the terms of the expres- 
sion are alike or similar. 

6. Similar Terms are terms which differ in their coefficients 
only, e.g., 5 • x, 6 • x, a • x, b • x. 

* That is, a number is exactly divisible by each of its factors. 



THE NUMBER SYSTEM 3 

7. Any factor (§ 2) of a number is the coefficient of the 
remaining factors. 

Thus, in a • sc, a is the coefficient of a;, 
in 2 • 3, 2 is the coefficient of 3. 
in 2 • 3, 3 is the coefficient of 2. 
in 2 • a • 6, 2 • b is the coefficient of a. 
in 2 • a • 6, a is the coefficient of 2 • 6. 
in 2 • a • 6, 2 is the numerical coefficient of a • 6. 
in ax, a is the literal coefficient of x. 

When the product of a number of figures and letters is to be 
written, the multiplication sign is usually omitted. 
Thus, 2 • a • b is written 2 ab. 

8. 2 a + 3 a + 7 a is a trinomial consisting of similar terms 
(§ 6). These terms may be united into one term by finding 
the sum of the coefficients. 

Hence, 2 a + 3 a + 7 a = (2 + 3 + 7) a = 12 a. 

This. is the same operation as that in arithmetic when one finds 

the value of 

2 ft. + 3 ft. + 7 ft. = 12 ft., 

and is brought still closer to arithmetic when one remembers 
that only like numbers can be added. 

Similarly, 15 ab — 3 ab + 7 ab 

means that 3 ab is to be subtracted from 15 ab, and 7 ab added 
to this difference. 

Hence, 15 ab — 3 ab + 7 ab = 19 ab. 

Ex. 1. Find the sum of 20 xy + 4 xy — 7 d. 

20 xy + 4xy — 7 d= (20 ay + 4 xy) —7d = 2±xy — 7d. 
Ex. 2. Add 5 a 2 + 3 a& + 4 6 2 and - 4 a 2 - 2 a& - 4 6 2 . 

_ 4 q2 _ 2 a & - 4 52 
a 2 + a& 



4 MATHEMATICS 

EXERCISE 2 
Find the sum of the following . 

1. 21# + 9a? — 4# + 3x — 8x. 

21a + 9# = 30# 
30x-±x = 26x 
26x + 3x = 29x 
29z-8a = 21a. 

This work is all to be done mentally, only results of each 
addition being given. 

2. 5m — 4:m + 6m — 2m. 

3. 8xy + 3xy — 2d. 

4. 8 a + 4 b, 4 a - 2 6. 

5. 16 a 2 + 8 a& + 5 b 2 , 5 a 2 - 3 ab + 2 6 2 . 

6. 21x 2 + 22xy + 17y 2 , - 8x 2 + 2xy - 9y 2 , 

and 7 a; 2 — 11 xy — 7 y 2 . 

7. 24a 2 +48a& + 246 2 , - 23a 2 - 47 a£ + 236*, 

and a 2 + 2 ab - b 2 . 

8. 14 c 2 + 21 cd + 10 d 2 , - 9 c 2 - 12 c<2 - d 2 , 

and — 5 c 2 — 9 cd — 9 d 2 . 

9. 3-19 + 2.19-4-5.19. 

10. 3 • 27 + 2 . 27 - 4 • 27. 

11. 14 • 18 + 25 • 18 - 16 • 18 - 12 . 18. 

12. 41 • 63 - 27 • 63 - 12 . 63. 

13. Express 27 as a binomial. 

14. If x is the digit in tens' place and y in units' place, 
express the number as a binomial. 

15. Express 47 as a binomiaL 

16. Express 648 as a trinomial. 



THE NUMBER SYSTEM 5 

17. If hundreds' digit is x, tens' digit y, units' digit z, 
express the number as a trinomial. 

18. If the digits of example 17 are reversed, express the 
number. 

9. We have considered the decimal phase of our number 
system ; the prime factors are of equal importance. 

The prime factors of 15 are 3 and 5. 

45 = 3 • 3 • 5 or 3 2 • 5, where the 2 indicates the number of 
times 3 occurs as a factor. 

EXERCISE 3 

1. Learn the following squares : 

1 2 , 2 2 , 3 2 , 4 2 , 5 2 , 6 2 , 7 2 , ...30 2 . 

2. Learn the following cubes : 

1 3 , 2 3 , 3 3 , 4 3 , 5 3 , 6 3 , 7 3 , 8 3 , 9 3 , 10 3 , 11 3 ; 12 3 . 

10. Literal monomials may be separated into factors. 

Thus, a 3 = a • a - a. 

a 2 b =a • a • &. 
(ab)(ab) (ab) = (a6) 8 = a 3 6 3 . 
Similarly, (2 . 3) (2 . 3) (2 . 3) = (2 • 3) 3 = 2 3 . 3 3 = 6 3 . 

Ex. Find the prime factors of 225. 
225 = (15) 2 = (3.5) 2 = 3 2 .5 2 . 

EXERCISE 4 
Find the prime factors of the following : 

1. 18,27,24. 6. (60) 2 . 

2. (18) 2 ,(22) 2 . 7. 484. 

3. 361,520. 8. (36) 2 . 

4. 9 3 , (27) 2 , 729. 9. 625. 

5. (12) 2 , (12) 3 . 10. 225 • 72. 



6 MATHEMATICS 

11. An expression that is a factor of each of two or more 
expressions is said to be a common factor of them. 

Thus, 15 a 2 b = 3 . 5 . a . a > b. 

25a*b = &.a-a-b.. 
10 a 3 c = 2 . 5 • a • a • a • c- 

5, a, and a are the factors common to each of the numbers 15 a 2 b t 
25 a 2 6, 10 a 3 c 

EXERCISE 5 
Find the factors common to the following : 

1. 144, 729. 6. 84 s 8 , 54 ate, 9. 

2. 225, 5 4 . 7. 3125 a 5 , 625 a 4 , 125 a 3 . 

3. (2 a) 3 , 24 a 2 c. 8. 243 a 5 b 10 , 162 a 4 6, 135 6 3 . 

4. 75 # 2 y, 45 #?/ 2 . 9. a 4 b s c% a s b 4 cz\ ab 5 cx r . 

5. 361 a 4 6 2 , 38a 2 6, 114 a 8 6 3 . 10. 75 c% 125 cd\ 224 ad. 

12. What does a 2 ?/ mean ? How many factors are there in 
the expression ? In x 2 y, let x = 3, y = 5. What is the result ? 

EXERCISE 6 

In the following expressions substitute x = 1, ?/ = 0, z = 5, 
a = 3, 6 = 4, and compute the value of the result : 

1. 3 a 2 W = 3 • 3 2 • 4 3 = 3 - 9 - 64 = 1728. 

2. 2a#+4/ + 7a; 2 z = 2.1.0 + 4.0 2 _t-7.r.5 







= 


+ +35 






= 35. 




3. 


ax 2 z. 




9. 1 


4. 


35a + 3 2/ _6z. 




z 


5. 


10834 ayz. 




10. - ,/ + 5a; . 


6. 


(5 a + 2 6)a?. 




2 


7. 


2 a(3 6 + 4 2). 




11. (y + 5a)(y + 5«> 


8. 


7 z(2 a + 5 y). 




12. (6— a)Xi 



THE NUMBER SYSTEM 7 



13 . (p __ a + y)x. 15. (3 a + 2 b + 2 z) 2 . 

14. A=-S. 16 . (15 a _ 3 b _ 6 2)3 # 

13. The parenthesis as used in these examples denotes that 
the quantities inclosed are each subject to the same operation. 

Thus, (x + y)z means that the sum of x and y is to be multi- 
plied by z, or that both x and y are to be multiplied by z and 
the sum of the products taken. 

(2 +3) 2 indicates that the sum of 2 and 3 is to be "squared," 
i.e., used twice as a factor. 

(a + b) + (c + d) is read : the sum of c and d is to be added 
to the sum of a and b, or the sum of a and b plus the sum of 
c and d. 

Likewise, (a + b) — (c + d) indicates that the sum of c and d 
is to be subtracted from the sum of a and b. 

The forms of parenthesis are ( ), the frrace j \, the bracket [ ], 
and the vinculum . The vinculum is seldom used. 



EXERCISE 7 

1. What does 12 x 2 mean ? 

2. What does (12 a) 2 mean? 

3. What does 12 (x) 2 mean ? 

Perform the indicated operations : 

4. (18 a +12 a) + (5 a+ 2 a). 9. 5(a + 3a) - 2(a + 2a). 

5. (21a+2a)-(6a+3a). 10. 4[5(a+6a)]. 

6. (21a-2a) + (6a-3a). 11. 6 [2(5 a-3 a?)] -2 [4(2 a- a)]. 

7. (21a-2a)-(6a-3a). 12. (5 a + 18y) - (2 y + 3y). 

8. 3(6a-2)+4(2a?-l). 13. (8 x + 12y + 15 z) -, (x + 2 y). 



8 MATHEMATICS 

Oral Review 

The area of a rectangle equals the product of the base and 
altitude (length and breadth). Find the areas of the following 
rectangles : 



Length 


Breadth 




Length 


Breadth 


l. *17" 


6" 




10. 


19' 


15' 


17- 6 = (10 + 7)6 = 


60 + 42 


11. 


21' 


24' 


= 102. 






12. 


17' 


18' 


2. 18' 


6' 




13. 


15' 


13' 


3. 18" 


7" 




14. 


12" 


24" 


4. 19' 


V 




15. 


32" 


21" 


5. 20' 


9' 




16. 


19' 


28' 


6. 22' 


• 8' 




17. 


31' 


22' 


7. 15" 


12" 




18. 


106" 


12" 


15.- 12 = 15 (10 + 2) = 


: 150 +30 


19. 


115" 


12" 


= 180. 






20. 


106' 


21' 


8. 16' 


12' 




21. 


112' 


15' 


9. 18" 


12" 




22. 


1024' 


4' 



The area of a triangle is equal to one half the product of the 
base by the altitude. Find the areas of the following tri- 



angles : 














Base 


Altitude 


Base 


Altitude 




Base 


Altitude 


1. 16 


10 


6. 36 


18 


11. 


37 


19 


2. 14 


7 


7. 44 


22 


12. 


18 


25 


3. 24 


8 


8. 46 


83 


13. 


25 


16 


4. 32 


10 


9. 48 


24 


14. 


32 


16 


5. 24 


12 


10. 21 


15 


15. 


17 


21 



indicates inches, ' indicates feet. 



THE NUMBER SYSTEM 9 

The area of a circle is equal to 3.1416 (Abbreviation is ir, 
pronounced pi) times the square of the radius. 
Indicate the areas of the following circles : 

(D represents diameter ; B, radius ; A, area.) 

16. D = 12, A = 7T • 6 2 = 36 7T. 22. B = 27, A = 

17. 72 = 18, ^4 = 7r-18 2 = 3247T. 23. D = 28, ^L = 

18. B = 20,A= 24. D = 56, J. = 

19. D = 38, A= 25. D = 58,A = 

20. D = 46, J.= 26. i2 = 26, ^4 = 

21. i? = 17, A = 27. B = 16, .4 = 

Find the side of a square whose area is : 

28. 729 . 30. 900 32. 361 

29. 841 31. 441 33. 529 

Find the edge of a cube whose volume is : 

34. 729 36. 64 38. 343 

35. 512 37. 1331 39. 216 

The area of a trapezoid is equal to the product of one half 
the sum of the lower base (B), and the upper base (b), by the 
altitude (a). 

Find the areas of the following trapezoids : 





B 


b 


a 


40. 


10 


5 


6 




B 


b 


a 


41. 


18 


10 


4 


42. 


22 


16 


8 


43. 


24 


12 


15 


44. 


13 


13 


16 


45. 


14 


14 


5 


46. 


9 


18 


8 





B 


b 


a 


47. 


18 


8 


8 


48. 


16 


14 


9 


49. 


15 


13 


14 


50. 


24 


15 


6 


51. 


22 


11 


8 


52. 


15 


17 


17 



Is there anything unusual about the trapezoids in examples 
44 and 45? 



CHAPTER II 

Equations 

14. An equation is a statement that two quantities are equal. 
The sign of equality is =. 

Thus, x + 3 = 5 is read x plus 3 is equal to 5. 

15. To solve an equation is to find the value or values of 
some letter involved in the equation which will satisfy the 
given equation. 

16. When a number substituted for some letter in an equa- 
tion makes the sides of the equation identical, the equation is 
said to be satisfied. 

A number which satisfies an equation is called a root of the 
equation. The number is also said to be a solution of the equation. 

Ex. a + 3 = 5. (1) 

Substitute x = 2 in the equation. 
Then, 2 + 3 = 5. 

The two sides or members of the equation are the same or 
identical. 

The number on the left of the sign of equality is called the 
first member or side. The number on the right is the second 
member or side. Thus, in x + 3 = 5, x + 3 is the first member, 
and 5 is the second member. 

17. The kinds of equations that concern us at present are: 
The equation of condition. 

The identical equation or identity. 
The geometric equation. 

18. An equation of condition is an equation that is satisfied 
only by a definite set of values. 

10 



EQUATIONS 11 

E.g., in x + 3 = 5,x = 2 is the only value which can be found 
for x, which is a root of the equation. x + 3 = 5 is therefore 
an equation of condition, the condition being that a; must equal 2. 

19. An identity is an equation which is always true for any 
specified values of the letters involved in it. 

E.g., 2 a = a + a is true for any finite value of a. 

20. The geometric equation is an equation of two geometric 
figures. In general, the algebraic equation (§§ 18,19) is assumed 
to be true, and if its roots satisfy it the statement of equality 
is verified. The geometric equation must usually be proved 
to be true (§ 70). 

21. The operations used in equations are largely those of 
addition, subtraction, multiplication, and division. 

22. The laws governing the use of these operations are a set 
of statements assumed to be true, and known as axioms. 

23. The Axioms 

1. If the same number, or equal numbers, be added to equal 
numbers, the resulting numbers will be equal, 

2. If the same number, or equal numbers, be subtracted from 
equal numbers, the resulting numbers will be equal. 

3. If equal numbers be multiplied by the same number, or 
equal numbers, the resulting numbers will be equal. 

4. If equal numbers be divided by the same number, or equal 
numbers, the resulting numbers will be equal. It is not allowable 
to divide by 0. 

5. Any number equals itself 

6. Any number equals the sum of all its parts. 

7. Any number is greater than any of its parts. 

8. Two numbers which are equal to the same number, or to 
equal numbers, are equal. 



12 MATHEMATICS 

The Use of Axioms 

24. Ex. 1. Solve 2 x + 3 = 9. 

Subtract 3 from each member of the equation (Ax. 2). 

2z = 9-3. 
or 2x = 6. 

Divide each member by 2 (the coefficient of x) (Ax. 4). 

Then, x = 3. 

To verify this root, substitute 3 for x in the given equation. 

2-3 + 3 = 9. 
The equation is satisfied, hence 3 is a root. 

Ex.2. Solve y+H=ll + ^. 

* 6 3 2 

Multiply both members of the equation by 6, the L. C. M. of the 
denominators (Ax. 3). 

or 6 2/+ 11 = 42/ + 15. 

Subtract 4y from each side (Ax. 2). 
Then, 2?/+ 11 =15. 

Subtract 11 from each side (Ax. 2). 
Then, 2 y = 4. 

Divide both sides by 2 (Ax. 4) . 
And, y = 2. 

Verify by substituting y = 2 in the <7i'ue?i equation. 

2+n=M. 

6 3 2 

Simplify each member. 

12 + U = 8 + 15 
6 6 

Hence, 2 is a root of the equation. 

EXERCISE 8 
Solve the following equations and verify each root: 
l. 2y + 7y + 3 = 12. 2. 5a> + 7 = 3a + rr. 



EQUATIONS 13 



^-1^ = 3-2A- 5 - 4a; + 2 =6. (Ax.) 



3 - T- 



6. 8z + 2-3z = 4z + 4. 
6. 



4. 6x + -=2x + ~- 7 6m + 4 = 3m + 3 

' 2 + 4 2 ' 

V+ 5 + 4 3 + 20 
10. ^ + ^ + 4 = 11. 

__ 7a 5a 3 a . r> 

11. = r- £. 

4 10 4 



12. 
13. 



6a , 8a_,6_5a , y 19 

2 m + 9 5m _ 3 m .. ,, 
3 + 2 = 2 ' 

_. ±R 6R , 8R 3R . Q 

14 - -T-2i + u=^ + 3 - 

15. ? + f_5 + l = ? + 3i. 

3 2 4 6 3 6 

_ -»• X . O X X OH jr>."«^ 

16. — = 4-^- -4- 

8^4 2 4 8 + 12 

17. * + ?+? = 13. 

9. T 3 A 



18 . E + ^ = 21. 

7 14 7 

19. ?^ + ^ + 7 = a; + 14. 

3 • 9 n 



3 2 

"8 



O X . X O X q i 

20 - -l" + o-^- = 3 i- 



14 MATHEMATICS 

21. The sum of two numbers is 9, and one is twice the 
other. Find the numbers. 

Let x = the smaller number 

and 2 x = the greater number. 

2 x + x = the sum of the numbers. 
9 = the sum of the numbers. 
Then, 2x + x = 9 (Ax. 8), 

or 3 x = 9. 

x = 3, the smaller number. 
2 x = 6, the greater number. 

22. The difference between two numbers is 24, and the 
greater is four times the lesser. Find the numbers. 

23. The sum of two numbers is 48, and the greater is 
four more than the lesser. Find the numbers. 

24. A number is composed of two digits. The tens' digit is 
four times as great as units' digit, and the sum of the digits 
is 10. Find the number. (Exercise 2, Ex. 14.) 

25. A number is composed of two digits. The tens' digit is 
three times the units' digit, and the number is 54 more than 
the sum of the digits. Find the number. 

26. The distance around a rectangle is 120'. The length of the 
rectangle is 10' more than the breadth. Find the dimensions. 

q 27. Two rectangles each have an alti- 

tude of 8', the sum of their areas is 256, 
v and the difference of their lengths is 12'. 

\ Find length of each. 

\ 28. In a triangle ABC the area is 24, the 

yi " b altitude (a) is 8. Find the base (b). 

29. Each of two rectangles has an altitude of 12'. One base 
is 10' more than the other, and the sum of the bases is 25'. 
Find area of each. 

30. The sum of the bases of two triangles of equal altitude 
is 28'. One base is 9' more than the other. The area of the 
smaller is GO square feet. Find area of larger. 



EQUATIONS 15 

31. In a trapezoid ACDE, the area is 120 ; the lower base B is 
four times the upper base b ; the altitude is 8. Find the bases. 

32. The sum of the angles of a triangle is equal 

to 180°. The angle at A is 20° more than the R 

angle at B, and 50° more than the angle at C. / \ 
Find the angles. 

33. In the triangle ABC, Z A is twice Z B, and Z C is equal 
to the sum of Z A and Z 12. Find the angles. 

34. The sum of two numbers is 48 and one number is four 
times the other. Find the numbers. (Solve mentally.) 

35. If one half of a number, plus one third of the number, 
plus one twelfth of the number equals eleven, what is the 
number ? 

3.6. Three fifths of a number, plus five thirds of the number, 
minus six fifths of the number equals 4 T "t. What is the num- 
ber? 

37. A father has a certain amount of money and he gives 
half of it to his son. Later he gives him half of what he has 
left and then takes back half of all he has given him, leaving 
the boy $ 1.87^-. How much money did the father have at 
first? * 

In the following examples determine whether they are iden- 
tities or conditional equations, by substituting values for x. 
If you find more than two values that satisfy the equation, it 
is an identity. 

38. x*-6x + 9=(x-3)(x-3). 

39. x 2 + 12x = x(x4-12). 

40. a 2 - 9 = + 3)0-3). 

41. x 2 -10x + 25 = x 2 + 8x + 16. 

42. (x + 7)0 - 3) = x (x + 4) - 21. 

43. x 2 - 8 x - 20 = x (x - 8) - 20. 

44. x (x - 8) - 20 = (x - 10)O + 2 )- 



16 MATHEMATICS 

45. 4:X 2 -9x + 7 = Sx 2 -x-S. 

46. 4 x 2 - 12 x + 9 = (2 aj - 3) (2 x - 3). 

47. 6z 2 -2a+4 = 2a 2 + 10a-5. 

48. 5# 2 + 6a;+4 = (5a — 2)(a? + 4). 

49. x 2 - 12 aj + 36 = (x - 6) 2 . 

50. a; 2 + 36a; + 81 =(aj + 9) (a? + 9). 



CHAPTER III 

Positive and Negative Numbers. The Four Fundamental 

Operations 

25. Positive and Negative Numbers. In addition to the num- 
bers thus far used in computation, it is necessary in algebra to 
extend the idea of number somewhat farther. In many prob- 
lems the numbers involved seem to have an opposite sense. 
For example: If a man has $500 and owes $300, the $300 
opposes the $ 500. If a man is walking east, the opposite 
direction to that in which he is going is west. North and south 
are opposites. Temperatures above and below zero are oppo- 
sites. To express this sense of opposition, positive ( + ) and 
negative (— ) signs are used in mathematics. E.g., if toward 
the right is + , then toward the left is — . If north is con- 
sidered positive, south is negative. If assets are +, liabilities 
are — . 

exercise 9 

AC D B 1. If D is four units to 

the right of 0, and C two 
units to the left of 0, how far is it from C to D ? Does that 
mean to you the difference between the positions of point O 
and point D? If A is —3 units from 0, and B is +5 units 
from 0, what is the distance from A to B ? 

2. A man has $ 600 and owes $ 300. How much is he 
worth ? 

3. A man has $ 500 and owes $ 700. How much is he worth ? 

4. A man has $500 and owes $500. How much is he 
worth ? 

17 



18 MATHEMATICS 

5. Where on the Cleveland- Wooster railway line is a place 
— 10 miles north of Cleveland ? 

6. A man goes 5 miles north of Cleveland, then 9 miles 
south. How many miles north of Cleveland is he ? How 
many miles has he traveled ? Draw a diagram showing his 
route and his last position. 

7. If to the right is positive, measure + 6" from a point A. 
Call this point B. Measure — 9" from B. Call this point D. 
Where is D with respect to A ? 

8. Translate into English (a 2 — b 2 ) s . 

Write in symbols : The square of the result of the quo- 
tient of the sum of (a) and (6) by 2. 

9. The temperature at 6.00 a.m. is + 14° and during the 
morning it grows colder at the rate of 4° an hour. Required 
the temperatures at 9 a.m., at 10 a.m., and at noon. 

10. Find the numerical value of the following, when a = 3, 
6 = 5, c = 2, d = 4 : a*b_atf 

2c 2 U 2 ' 

11. Translate into English f^L+l\ 

Write in symbols : the cube of the result of subtracting 
the square of b from the square of a. 

12. Find the numerical value of the following, when x = \, 
y = ±,z = \: 4x*+(3y-2zy. 

Addition 

26. In §8, we found the sums of similar terms, but in each 
instance the sum was positive. A negative sum may arise from 
the addition of a positive and a negative number. For instance, 
in example 6, exercise 9, we are adding — 9 miles to + 5 miles. 
The result is — 4 miles. That is, the addition of a negative 
number to a iiositive number tends to lessen the numerical value 



FOUR FUNDAMENTAL OPERATIONS 19 

of the sum, annul it, or change it from positive to negative. If a 
man has $400 and owes $400, the sum of his assets and liabil- 
ities is 0. If he has $400 and owes $700, the sum of his 
assets and liabilities is — $ 300. That is, he owes $ 300 more 
than he has assets. The sum of two negative numbers is nega- 
tive. It is seen in these results that when the sum of a positive 
and a negative number is found, the result takes the sign of 
the greater absolute value. 

27. The absolute value of a number is its value regardless of 
sign. For example, the absolute value of — 6 is 6. 

28. If no sign is placed before a number it is regarded as 
positive. A negative sign must never be omitted. 

EXERCISE 10 

Find the sums of the following : 

1. +5 2. +5 3.-5 4.-5 

+3 -3 +3 -3 

5. 3 a 2 and — 4 a 2 . 

6. 7 ab, — 5 ab. 

7. — 8 a, — 5 a. 

8. 9« 2 , -6 a 2 , - 3 a 2 , - 12 a 2 , - 5 a 2 . 

9. 3 a, — 5 a, + 6 a, — 4 a. 
*10. 7 a 2 , 4.x, +3 a 2 , + 2 x, + x. 

11. 12a?y, <6xy 2 , ±x 2 y, -2x 2 y, + 11 xy 2 . 

12. 15a 3 , - 15a 2 , + 15x, + 5x 2 , - 5x, - 9x>, - 9a 2 , - 9a. 

13. 4:X + 3y— z, — 2a + ?/ + 4z, — a — 3y — 2z. 

14. 5 (a + 6), - 2 (a + 6), 6 (a + b), - 9 (a + V). 

* Write similar terms (§ 6) in the same column. Make as many columns as 
you have different kinds of terms, forming the whole into one problem, by 
usiug + and — signs. (See example 2, § 8.) 



20 MATHEMATICS 

15. 4 (x + y) - 3 (x - 2 y), - 3 (x + y) + 2 (x - 2 y), 

2(x + y) + (x-2y). 

16. 2 (a + ft) 2 - 6 (a + 6) + 1, - 5 (a + b) 2 + 3 (a + b) - 6, 
(a + bf - (a + 6) + 2. 

17. 2 a,- I?, + i 2, |aj + | y _^ __ |aj + ^y-,i 2 . 

18. fa- fft + fc, T \a-i&-fc, -fa + Jft-^c. 

Find the value of the following sums when x = ^, y = + \ } 
z = ±,a = -2,c = ±,b = 2: 

19. | a + i 6 — c, +a — \b — %c, 5a — f 6 + 2 c. 

20. 2(a + 6) 2 - 3(6 - c) 2 + (a - 6), - 500(a + b) 2 + 5(6 - c) 2 
+ (a - 6). 

21. 5xy — 5 x 2 y — 5 a??/ 2 , \ xy + -| a 2 ?/. 

22. 7(a + 2y)-7(a>-2y), - 31(s + 2y) + f(s- 2y). 

23. 12yz — 8xy + \a + %bc. 

24. f a — f 6 + f c, +f a--i-6 + 25c. 

In each of the following examples, add corresponding mem- 
bers of the two equations to find x : 

25. x + y = 8, 

x — y = 4:. When # is found, can you find y ? 

26. 2a? + 3y = 8, 

# — 3y — — 5 . Find y also. 

27. 4z-2?/= 2, 

3 a: + 2 y = 12 . Find y also. 

28. Verify your results in examples 25-27 by substituting 
the values found for x and y in the given equations. 

Subtraction 

29. (a) What number added to 9 gives 7 ? 
(b) What number added to 9 gives 11 ? 



FOUR FUNDAMENTAL OPERATIONS 21 

(c) What number added to 9 gives ? 

(d) What number added to 9 gives — 12 ? 

In each of the above examples we have the sum of two num- 
bers and one of the numbers given to find the other number. 

The Minuend is the sum of two numbers. 

The Subtrahend is a given number. 

The Difference is a required number when the minuend and 
subtrahend are known. 

Subtraction is the process of finding what number added to 
the subtrahend produces the minuend. 

In example (a), 7 is the minuend, 9 is the subtrahend, — 2 
the difference. In subtraction, the sum of the subtrahend and 
the difference must equal the minuend. This fact enables us 
to check our result. 

EXERCISE 11 

1. Subtract — 3 from 5. 

Here 5 is the sum of the numbers. — 3 is one of the numbers. Our 
problem is : What number added to — 3 gives 5. This number is evi- 
dently 8. That is, the difference is 8. 

Check : 8 + (— 3) = sum of the numbers. 

5 = sum of the numbers. 

The written work stands : 

5 

-3 

8 



2. From — 8 a take 5 a : 



- 8a 
+ 5a 



-13 a 

Perform the following subtractions : 

3. 7x 2 y 4. 25 x 5. 13 x 6. — 25x 7. +25x 
-3arV 13a 25x -13x -13 a 



22 MATHEMATICS 

8. From — 13 ab take 24 ab. 

9. From a + b + c take a — b — 2 c. 

(Note that there are three subtraction examples in this example, one 
for each column.) 

10. Subtract 8x — 2y + z from 10 x — y — 3 z. 

11. Subtract 2 a 2 — 3 a + 1 from 5 a 2 — 3 a — 1. 

12. Subtract 10 a 2 + 5 ab - 9 6 2 from 2 a 2 - 10 a& + 8 b\ 

13. From x s + 3 a 2 -f- 3 x + 1 take a 8 + 2 a? 2 + 2 a; + 1. 

14. From x* + 3 a 2 + 3 a? + 1 take x 3 - 3 a? 2 + 3 x - 1. 

15. From a 2 + 2ab + b 2 take a 2 - 2 ab + & 2 . 

16. From 5 ir 2 take 5 ?/ 2 . 

17. From 6 a 2 - 5 x + 4 take 12 o; 2 - 9. 

18. From 3x 2 take - 4.x 3 + 3x 2 - 2x + l. 

19. Subtract a from 0. 

20. Subtract 9 x 2 + 9 y + 9 from 0. 

Note that in each of the above examples the difference is 
the same as if we had changed the sign of the subtrahend and 
added the result to the minuend. 

We may then use the following rule for subtraction. Change 
the sign of each term of the subtrahend and proceed as in addi- 
tion, {The change of sign must be made mentally.) 

EXERCISE 12 

1. What must be added to 9 x 2 + 9 ?/ + 9 to give ? 

2. From x 2 + 4 x + 4 take x 2 — 4 x + 4. 

3. From the sum of a 2 + 2ab + b 2 and a 2 - 2a6 + ft a take 
the difference between a 2 + 2 ab + b 2 and a 2 — 2 ab + b' 2 * 

*In this text the difference between a and b means the remainder obtained 
by subtracting b from a. 



FOUR FUNDAMENTAL OPERATIONS 23 

4. Subtract 2 a s + a 2 b + ab 2 + 2 fr 3 from the sum of a 3 + 3 a 2 & 
+ 3 ab 2 + 6 3 and a 3 - 3 a 2 & + 3 ab 2 - b\ 

5. Subtract x 2 + 3 x — 4 from ft 3 . 

6. Subtract 6 + c from a. 

7. Subtract 4 ft 2 - 7 a - 9 from 0. 

8. What shall we add to 7 ft 2 — 12 ft + 5 to produce ? 

9. [5 ft 3 - (3 x + 2)] - [6 a; 3 + (4 ft -11)]. 

In examples of this tvpe perform the indicated operations 
one at a time, beginning with the inner parenthesis. First sub- 
tract 3 ft + 2 from 5 ft 3 , then add 4 x — 11 to 6 ft 3 . This gives 

[5^_3ft-2]-[6ft 3 + 4 ft-11]. 

Subtracting the second trinomial from the first, we have 

Translate each example into English before solving : 

10. {5a— [2 a + (4 a + 1)]|. 

11. [12ft 2 - {-7ft 2 - (5 ft 2 + 6 ft- 3)}]. 

12. [6 (5 a + 3) + 5 (2 a + 7)] -(6 a + 53). 

13. [25 m - (2 m + 3)] - [- 10 m - (6 m - 7)]. 

14. [ft 3 - (3 ft 2 ?/ - 3 xif - f)~\ - [ft 3 + (- 3 x 2 y + 3 ft?/ 2 - 1/ 3 )]. 

15. (12 a + 3 6 + 2 c) - (5 a - 2 b + 6 c) - (10 a - b - 6 c). 

16. 4(* + y)-[6(aj-y)] + (a?-2y). 

17. There are two numbers, ft and y, whose sum is 17 and 
whose difference is 1, x being the greater number. Find the 
numbers. 



By the conditions, 




x + y = 17 


(1) 






x-y = 1 


(2) 


Adding (1) and (2), 




2x = 18 
x = 9. 




Subtracting (2) from 


(1), 


2 y = 16. 

y = 8. 





24 MATHEMATICS 

18. Given two numbers, x and y, such that the second num- 
ber added to twice the first is equal to 12, and the difference 
between twice the first number and the second number is 8. 
Find the numbers. 

19. 5x + 2y = 19, (1) 

5x — 2y = ll . (2) Find x and y. 

20. x + 3 y = 16, (1) 
a?-3y=-14 (2) 

21. 2x +3y = 17, (1) 

a? — 3.y= — 14. (2) After x is found, obtain the 

value for y by substituting the value of x in equation (1). 

22. Find the value of the difference between 5 a?+ 3 x?y—5a 2 
and 2 a? — 3 x 2 y — 2 a 2 , when x = 5, y = %, a = 3. 

Multiplication 
30. In addition we learned that 

5 + 5 + 5 + 5 = 5.4 = 20 
Also that 

(- 5) + (- 5) + (- 5) + (- 5) = (- 5)(4) = - 20. 

This last shows that the product of a negative number by 
a positive number is negative. 

In arithmetic, it does not matter in what order the factors 
are used, e.g., 5-3 = 3-5. 

We shall assume that this law holds true in algebra. 

Then, (-5) (4) =(4) (-5)= -20. 

That is, the product of a positive number by a negative number 
is negative. It seems then that multiplication by a negative 
number gives to the product a sign opposite to that of the 
multiplicand. 

Ex. (-8). (-5)= +40. 



FOUR FUNDAMENTAL OPERATIONS 25 

31. These rules for signs follow: 

The product of two numbers of like sign is positive. 
The product of two numbers of unlike sign is negative. 

32. Since division is the inverse of multiplication, the same 
sign rules hold ; namely, 

When the terms have like signs, the quotient is positive; when 
unlike, the quotient is negative. 

Division may be regarded as an application of the method 
of subtraction. 

Ex. Divide 28 by 7. 

This is equivalent to successively subtracting 7 until no remainder or a 
remainder less than 7 remains. 

28-7 = 21, 21-7=14, 14-7 = 7, 7-7=0. 

Which shows that 7 is contained in 28 four times without a remainder, 
or that 7 is an integral factor of 28. 

Regarding division as the inverse of multiplication is the 
more general method. With this understanding we have the 
following definitions. 

33. Division is the process of finding one of two factors 
when their product and one of the factors are given. 

The Dividend is the product of the factors. 

The Divisor is the given factor. 

The Quotient is the required factor. 

It is evident from these definitions that the product of the 
divisor by the quotient is equal to the dividend. By the use 
of this principle the accuracy of the result of the division may 
be verified. 

Give sign rules for addition, subtraction, multiplication, and 
division. 



26 MATHEMATICS 

EXERCISE 13 

Find the product of the following (first determine the sign 
of the product) : 

1. (4)(-3). 4. (+10)(+10). 

2. (-9)(4). 5. (_10)(-10). 

3. (-9)(-8). 6. (-7)(+9). 

Find the following quotients : 

7. (+12) + (+4). 10. (-20)-f-(-4). 

8. (-12)-*- (-3). 11. (-112) -(16). 

9. (-15) -5- (+5). 12. (144) -s- (-18). 

13. The area of a rectangle is 212, the base is 17. Find the 
altitude. 

14. The area of a triangle is 180, the base is 20. Find the 
altitude. Can you construct the triangle ? Why ? 

15. The area of a rectangle is 24, the altitude is — 4. What 
is the base ? How do you account for these negative results ? 
Draw this rectangle. (§ 25.) 

34. In multiplication it is convenient to show the number of 
times a quantity is used as a factor by means of a symbol 
placed at the right and above a quantity. 

This symbol showing to what power the number is to be 
raised is the exponent of the number. 

Ex. 1. x • x • x = a?. (2 x)(2 x)(2 x)(2 x) = (2 x) 4 = 16 x\ 

Similarly, x • x • x ••• n factors = x n . 

Ex. 2. x 2 • x A = x • x • x • x • x • x = x e . 

And x n • x m = (x • x '• x ••• n factors) (x-x>x • • • m factors) 
= x-X'X--m + 7i factors = x m+n * 

Then in multiplying two like letters give to their product an 
exponent equal to the sum of their exponents in multiplicand 
and multiplier. 

* ... is read, " and so on to." 



FOUR FUNDAMENTAL OPERATIONS 27 

We see that multiplication is simply combining the factors 
of the multiplicand and multiplier into one product ; e.g., 12 • 18 
means (2 2 • 3) (3 2 • 2) or 2 3 . 3 3 , the product containing all the factors 
of both multiplicand and multiplier. Similarly, 24a 2 6-15a 3 

= 2 3 - 3 • a 2 • b ■ 3 • 5 . a 3 = 2 3 . 3 2 - 5 ■ a 5 - b = 360 a 5 b. 

35. Since division is the inverse of multiplication, the ex- 
ponent rule for division is the reverse of that in multiplication. 

In dividing a letter by the same letter having the same or a 
different exponent, give to the quotient an exponent equal to 
the exponent of the dividend minus the exponent of the divisor. 

Ex. 1. Divide X s by x 2 . 

X 2 = X • X. 

Then, (x • x • x) -4- (x • x) = £, 

or x 3 -T- x 2 = x s ~ 2 = x 1 . 

When the exponent is 1, it is not expressed. 
Thus, x s -r- x 2 — x. 

Ex. 2. Divide a 15 by a 12 . 

«15 ^ a 12 _ a 15-12 = a 3 # 

EXERCISE 14 

Find the following indicated products : 

(Translate each example into English before solving.) 

1. x 5 -x\ 8. (12 c 3 ) - (-4 c 5 ). 

2. 15 x 2 • 5 X s . 9. a s b • a 4 b 2 . 

3. 8m- 18 m\ 10. -15 a 2 6 2 • 21 a z b\ 

4. (15a) 3 (2a). 11. (- 13 a 2 b 6 c)(- 13). 

5. (a? + y) a (a5 + y)». 12. (18 c 3 d)(- 19 c A d 2 ). 

6. 12(a + 6) • 3(a + &). 13. (27 ab 2 xy)(- 14 a 5 6c). 

7. _19(a-6) 4 .15(a-6r. 14. (- 21 afy*)(- 21 afyV). 



28 MATHEMATICS 

15. 12(a + b) by 12(a + 5) 2 by 12(a + 6) 3 . What power of 
2 is in your product? What power of 3? What power of 
(a + b) ? What then are the prime factors of your product ? 

Indicate the prime factors of these products : 

16. (18a 3 6 4 )(18a& 2 ). 

17. ( - 24 p 2 g 4 r) ( - 2±p 2 g 4 r) . 

18. 32(a + x) 5 by 64(a + a) 6 . 

19. 27 (c + df by 9(c + d) 2 . 

• 20. 125(a - ?/) 3 by 25(a - ?/) 2 . 
Find these quotients and verify your results : 

21. x 8 -r-x 5 . 27. 729 cV -s- 9 ex 8 . 

22. x A +-x. 28. (c — d) i0 -r- (c — d)\ 

23. 72 x 6 -5- 9 x\ 29. 27 (a + b) s ~ 9(a + 6) 2 . 

24. 441 a 6 -4-21 a 3 . 30- a 5 b 3 c 2 -*- a 2 bc. 

25. (_24^ 3 )-(-3^ 2 ). 31. 2 5 .5 3 -3 2 -f-2 2 .5.3. 

26. (576a 7 6 4 )--(-24a 5 6 2 ). 

Are examples 30 and 31 similar ? 

32. Divide 5 4 . 3 5 . 2 4 a 7 6 V by 5 3 ■ 2 4 a 7 6 y. 

Factor the dividend and divisor. Then divide as in ex- 
ample 32 : 

33. 162 x A y ~ 54 xy- 34. - 135 cV ■*• 15 cx\ 

35. - 210 a^yV -*■ - 14 x 4 ?/ 3 . 

36. The area of a rectangle is 128 (a; + y)(c + d), its altitude 
is 8(c + cT). What is its base ? What are its dimensions if 
x = l, y = 2, d=4* c= -3? 

37. The area of a rectangle is 384 artyc 2 , its altitude is 16# 2 c. 
Find its base. What are its dimensions if aj = — 2, y = — 1, 
c=— 2? Also if a? = 2, y = l, c = 2? Draw these rec- 
tangles, measuring from the same starting point for each. 



FOUR FUNDAMENTAL OPERATIONS 29 

38. The base of a triangle is 27 a 3 c A , the area is 243 aV. 
Find the altitude. What are the dimensions when a = — 3, 
_ i? 



c = 



3 



36. Give the sign rules for addition, subtraction, multipli- 
cation, division. 

Give the exponent rules for multiplication and division. 

Supplemental Applied Mathematics 

A decimal fraction is a fraction whose denominator is a power 
of 10. This denominator may be written, or simply expressed 
by the relative position, of the decimal point. The factors of 
10 are 2 and 5. The factors of 100 are two 2's and two 5's, 
that is, 100 = 2 2 -5\ Likewise 1000 = 2 s -5 3 . Every power of 
10 is made up of an equal number of 2's and 5's used as fac- 
tors. Therefore, to reduce a common fraction to & pure decimal 
one must multiply both numerator and denominator by such 
number as will produce an equal number of 2 and 5 factors 
only in the denominator. 

Ex. 1. Reduce f to a decimal. 

q q Two 5's are lacking. Multiplying both numerator and 

- = - 2 - denominator by 5 2 or 25, we have -^ , or, expressed decimally, 
.75. 

Ex. 2. Reduce to .1875 a common fraction: 

.1875 = 1^ = ^- = A. 

2±-5* 2*-5 4 16 

3. Reduce to decimals : \ y f , ^, ±%, |, |f' 

4. A machinist has a set of drills marked .1250; .9375; 
.0625 ; .03125 ; .3125 ; .8750. He does not recognize them as 
readily as if they were in 8ths, 16ths, 32ds, 64ths. Reduce 
them to such notation. 

5. Reduce the following sizes of drills to their decimal 
equivalents : 1-8, 3-32, 3-16, 5-16, 7-16, 9-16, 11-16, 13-16, 
19-16. 



30 MATHEMATICS 

In all computation in science and shop practice the decimal 
plays an important part. 

Whether multiplication and division is carried on directly or 
by means of tables, the computer must know at a glance where 
the decimal point is to be placed. 

Multiplication of Decimals 

Decimals are multiplied as simple numbers if one remembers 
the decimal composition of our system and the part the posi- 
tion of the digit plays in the formation of a number. In 145, 
§ 1 and exercise 1, the 1 has 100 times the value it would have 
if written in units' column where the 5 now is. The 4 has ten 
times the value it would have if in units' column. Then, mul- 
tiplication by a digit in hundreds' column has 100 times the 
effect of multiplication by the same number in units' column. 
A similar statement holds for digits in tenths' and hundredths' 
columns, each move to the right decreasing the value of a digit 
ten times. 

In multiplication we begin at the left. 

Ex. 6. Multiply 24 by 36. 

24 

36 

720 
144 

864 

We multiply first by 3. Since this figure is in tens 1 column, it has ten 
times the value of a figure in units' column, and our product moves one 
place to the left. That is, we are really multiplying by 30 units. The 
units' column, not the decimal point, is the dividing line. 

Ex. 7. Multiply 23.2 by 2.4. 

23.2 
2.4 



46.4 

9.28 

65.68 



FOUR FUNDAMENTAL OPERATIONS 31 

Always keep the decimal points in the same vertical column. 
When multiplying by 4, the multiplier was one place to the 
right of units, and the product y 1 ^ as large as the product by the 
same figure if in units' place. This shifted our product, 9.28, 
one place to the right. 

Always begin multiplication at the left. 

Find the following products : 



8. 
9. 


17 • 26. 
324 ■ 14. 


17. 
18. 


(2.7) 2 . 
(.27) 2 . 


26. .6425 (.0125). 
How much multiplying 


10. 
11. 
12. 
13. 


324 • 324. 
216 • 36. 

9 2 - (3.1416). 
16 2 - (3.1416). 


19. 
20. 
21. 
22. 


(.09) 3 . 
24.21 (.32). 
1.875 . 16.2. 
1.875 • 1.62. 


is necessary in ex- 
ample 27, if the result 
is correct to three 
decimal places ? 


14. 


24-62.5. (.92). 


23. 


1.875 (.162). 


27. 4.261 (.7854). 


15. 
16. 


(.8)3. 
(1.2)3. 


24. 
25. 


18.75 (1.62). 
1.112 (.99). 


28. 32.15 (.625). 



Division of Decimals 

Here we must reverse our work of multiplication. The first 
figure of the divisor and its distance from units' column deter- 
mine the position of the decimal point with respect to the first 
figure of the quotient. If the first figure of the divisor is in 
tens' column, the quotient will be ten times as small (or one 
tenth as much) as if the divisor were units. The first figure of 
the quotient therefore moves one place to the right of the first 
figure of the dividend. 

Ex. 29. Divide 144. by 72. 

02. 

72. | 144. 

144. 



Note that 7 is not contained in 1, the first figure of the quotient; there- 
fore a zero is written for the first figure of the quotient. Since this zero 



32 



MATHEMATICS 



is to the left of an integer, it has no value and may be disregarded. Until 
the beginner has the placing of the first figure of the quotient well in mind, 
this zero should be employed. 



Ex. 30. Divide 324. by 18. 



18. 



18. 



324. 

18 . 



144. 
144. 



Ex. 31. Divide 2.446 by 3.2. 



0.76 + 



3.2 



2.446 
2.24 



.206 
.192 

.014 

Ex. 32. Divide 2.446 by .32. 

07.6 + 



32 



2.446 
2.24 



Ex. 33. Divide 2.446 by 32. 



.206 
.192 

.014 



.076 + 






32. 



2.446 
2.24 



.206 
.192 

.014 

Note that the position of the first figure of the divisor alone controls 
the position of the first figure of the quotient. 

Find the following quotients. (Use three decimal places if 
the quotient is not exact.) 



FOUR FUNDAMENTAL OPERATIONS 33 

34. 31.5 -=-24,6. 44. 251.328-5-3.1416. 

35. 6.4-5-1.6. 45. 442 -5- .09. 

36. 6.4 -f-. 16. 46. 2.24-5-22.4. 

37. 6.4 -.016. 47. 361. -s- .19. 

38. 6.4-5-16. 48. 5.29 -=- .23. 

39. .225-5-15. 49. 7.29-5-2.7. 

40. .225-5-1.5. 50. 7.29 -5- .27. 

41. .225 -5- .15. 51. .0289-5-1.7. 

42. 109.624 -5- 3.86. 52. .0289 -5- .17. 

43. 125.664-5-3.1416. 53. .0289-5-17. 

Locate by inspection the first figure in each of the following 
quotients. (Remember that the number of places in the quo- 
tient does not in any way depend upon the number of figures in 
the divisor.) The position of the first figure in the divisor is 
all that one needs consider. 

For example, in dividing 8.432694 by .3419768321, the first 
figure 2 takes the same position as if we were dividing by .3, 
namely, in tens' column. 

2 . 
.3419768321 | 8.432694 

54. 48.36579-5-4.6293251. 60. 34.76-5-38, 

55. 4.836579 -s- 4.6293251764. 61. .026947321 -- .41976384. 

56. 4.836 -s- 4.6293251764847. 62. .178643791 -5- 2.9. 

57. .2793 -5- .217398. 63. .243 -*- .0986432791. 

58. .2793 -5- .0217398. 64. 2.43 -5- .986432791. 

59. 31.84-5-309.7. 65. .0243-5-9.864327915. 

66. Multiply 16346.2' by .00019, and show that the product 
is in miles. 

67. Divide 3 cubic feet by .00058, and show that the quotient 
is cubic inches. 



34 MATHEMATICS 

68. Rolled oats requires If hours cooking on a range. If a 
tireless cooker is used, 15 minutes cooking on the range is suffi- 
cient. How much fuel is saved by using the tireless cooker, if 
8.6 cu. ft. of gas per hour are consumed by a gas burner, gas 
costing 76^ per thousand cubic feet ? 

69. Wheatena, or cream of wheat, requires f hours cooking 
on a range, or 15 minutes cooking on a range for a fireless 
cooker. How much fuel is saved by using the fireless cooker, 
gas burner and price same as in Ex. 68 ? 

70. Corn meal mush should be cooked for three hours on a 
range, or for 15 minutes if a tireless cooker is used. How 
much fuel is saved by using a tireless cooker ? 

71. Uncooked rice contains 79 % carbohydrates, while boiled 
rice contains 24.4 %. How much carbohydrate is lost from one 
pound of rice by boiling ? 

72. Eice boils in 20 minutes, using a gas burner consuming 
8.6 cubic feet per hour. Steamed rice is cooked on the same 
burner for 5 minutes, and on the simmering burner for 45 
minutes, the latter consuming 3.6 cubic feet per hour. What 
is the difference in the cost of cooking ? 

73. Rice swells 3f times by boiling. If a recipe for pudding 
calls for one quart boiled rice, how much uncooked rice should 
be used ? 

In the following equations solve for x and verify the root 
found : 

74. 2 x — 4 = 7 x + 16. 

75. 5x- $ + 2x = x -32. 

76. 2 x — a = 5 a. 

77. 2x — a = b. 

78. 3x — 2b + x — a = a + 2x. 






l-5 + l 

2 2 



79. ~ _5 + ; ,-=-2a> + 5$. 



FOUR FUNDAMENTAL OPERATIONS 35 

80. 4^-3^ + ^-1 = ^-2. 

8i. ?_10_2i a; = iX-14If + 2*. 
o o 

82. The length of a room is one and one half times its 
width. If three feet is taken from the length and three feet 
is added to the width, the room will be square. Find its di- 
mensions. 

83. The sum of one third, one fourth, and one fifth of a 
number is 17 more than one half the number. Find the 
number. 

84. The distance around a rectangle is 22-| yards. The 
length is 2\ yards more than the width. Find the square 
yards in the rectangle. 



CHAPTER IV 

Polynomials. Multiplication of Polynomials 

37. Polynomial by Monomial. Review multiplication of 
monomials, §§ 31-34. Give sign rule for multiplication. 
Give exponent rule for multiplication. Give sign rule for 
addition. Illustrate each. 

38. In § 34, we found the product of a monomial by a 
monomial. We shall now extend multiplication to cover any 
number of terms. 

A polynomial is simply a sum of monomial terms. 

Hence, to multiply a polynomial by a number is to multiply 
each of its terms by that number, and find the sum of these 
products. 

Ex. Multiply 5 a 2 + 3 ax — x 2 by 2 a. 

5 a 2 . 2 a = 10 a 3 , 3 ax • 2 a = 6 a 2 x, -x 2 .2a=-2 ax 2 . 

Then, (5 a 2 + 3 ax - x 2 ) - (2 a) = 10 a 3 + 6 a 2 x - 2 ax 2 . 

The work should be written in the following form : 

5 a 2 + 3 ax — x 2 
2a 



10 a 3 -f 6 a' 2 x — 2 ax 2 
Begin multiplication at the left. 

EXERCISE 15 

Find the following products. (Perform the numerical multi- 
plication mentally, writing results only.) 

36 



MULTIPLICATION OF POLYNOMIALS 37 

7a 2 + 3ab + 2b 2 3. -4.x 2 + 2xz-5z 2 

5a —6x 



2. 5 m 2 -3 cm + y 4. — 14 ad 2 + 15 a 2 d + 17 

2c 3 v -13a 



5. 18x 2 -17a3y-242/ 2 
12 a 4 

6. In example 5, substitute x = l, y = 2 in your multipli- 
cand, multiplier, and product. Is the result what you might 
expect? 

Any example in multiplication may be checked by substitut- 
ing some numerical value for each letter as suggested in exam- 
ple 6. 

7. Check each of the first five examples. 
Multiply : 

8. 32 a?z — 14 xz s by 16 xz. 

9. 115xy 2 -112x s y by -12. 

10. 1024 a 2 + 612ab- 306 b 2 by 4 ab 2 . 

11. ( x + y) 2 + 6(x + y) + 9 by (x + y). 

12. (x + yf + 3(x + y)-4, byl6(x + y) 2 . 

13. _24(£-2/) 3 + 14(a + &) 2 -21 by 16. 

14. 3(x + y) 2 + 12(x + y) + lS by±(x + y). 

15. Check example 14. 

39. Polynomial by Polynomial. To multiply a polynomial 
by a polynomial is to multiply the multiplicand by each term 
of the multiplier, and add the partial products. 

Ex. 1. Find the product of 2 a + 3 b bj 3 a — 5 b. 

(2 a + 3 6)3 a = 6 a 2 + 9 ab, (§ 37) 

(2 a + 3 b) (- 5 6) = - 10 ab - 15 & 2 . 



38 MATHEMATICS 

Adding these partial products 

6a 2 + 9ab 

- 10 ab - 15 b 2 
we have 6 a 2 - ab - 15 6 2 

The work should be written as follows : 

2a + 3b 
3a-5b 



6a 2 + 9ab 

- 10 ab - 15 & 2 
6 a 2 - a& - 15 b 2 

The procedure is the same for any number of terms. The 
pupil will find the work more simple if both multiplicand and 
multiplier are arranged according to either the descending or 
ascending powers of some letter. 

Ex.2. Multiply 5x- 6x 2 + x 3 - 4 by -3x + x 2 -l. 

Rearranging according to the descending powers of x, 

x 3 - 6 x 2 + 5 x - 4 

X 2 _ 3 x _ i 

x b — 6 x 4 4- 5 x s — 4 x 2 

-3x*+ 18x 3 -15x 2 4-12x 

— x 3 4- 6 x 2 — 5x + 4 

X 5 _ 9 X 4 + 22 x 3 - 13 x l + 7x + 4 

EXERCISE 16 

Find the following products. Check each result : 

1. (5a + 3b)(2a+±b). 6. (ox i -±x 2 +3x-2)(2x-7). 

2. (6a + 2y)(5a?-3y). 7. ( x 2 + x + 2)(x 2 -x + 2). 

3. (7 c + 2d) (7c -3d). 8. (a 2 + .«# + #*) (a 2 - xy + y*). 

4. (7c + 2d)(lc-2d). 9. (a 2 + 2aft + & 2 )( a + &)« 

5. (a^ + a; 2 + l)(x-l). 10. (a 2 - 2 a& + &*) (a - 6). 
Are the products in 9 and 10 alike ? 

11. Multiply a' 2 + 2 a + 4 by a - 2. 

12. Multiply 5 a 2 — 30 as 4- 45 by 5x — 15. 



MULTIPLICATION OF POLYNOMIALS 39 

13. (a + 6) (a + 6). 15. {x-2y)\ 

14. (a + 2b)(a + 2b). 16. (x-y)(x-y). 

Note the form of these results in examples 13-16. They 
will be useful later. 

17. (a + b)(a-b). 19. (4a + y)(4a>-y). 

18. (a + 26)(a-26). 20. (4a + 3y)(4a>-3y). 

21. (16c + 15d)(16c-15d). 

Note the form of the results in examples 17-21. 
Write the following products by inspection : 

22. (c + 2d)(c-2d). 25. (3x + 2y)(3x + 2y). 

23. (5x + y)(px-y). 26. (2 a + 3 c) (2a + 3 c). 

24. (5a + y)(5aj + y). 27. (5ra + 4fc)(ora + 4&). 

Solve examples 28-32 mentally : 

28. The side of a square is a -f 6. Find the area. 

29. The side of a square is 2a — b. Find the area. 

30. The side of a square is £x + 3y. Find the area. 

31. The side of a square is 6 c — 4 d\ Find the area. 

32. The side of a square is 5 #— 2 a. Find the area. 

33. In examples 28-32 find the dimensions and the area if 
a = 2, b = 1, c = — 1, d = 3, x = 4, y = — 1. 

Find the areas of the following A where 5 = base and a = 
altitude : 

b a 

34. 6# + 3?/, 5x + 7y. 

35. a + 5, a — &. 

36. x + S, x—2. 

37. a + 9 y, x + 2 y. 

38. x + 15 a, x — 3 a. 



40 MATHEMATICS 

39. Compute the areas in examples 34-38 when a = 3, 6 = 1, 
x = 7, y = - 6. 

40. 16 x* + (8 x 2 - 3 x + U)(5 x -2 x 2 + 24) - 336 = ? 

41. [(a? 2 + 3 a; - 4) + (x 2 - 3 x + 4)](9 a 2 - 6 a? + 1) = ? 

42. (2a + 3)(2a + 3) 2 =? 

43. The edge of a cube is 4 a + 6 ; find its volume. 

44. The edge of a cube is 2 x — 5 y\ find its volume. 

45. The edge of a cube is 5 x — 14 6 ; find its volume. 

46. Find the volumes in examples 43-45 if the letters have 
the same values as in example 33. 

Find the areas of the following trapezoids : (B represents 
the lower base, 6 the upper base, and a the altitude). 

B b a 

47. x + y, x-y, 2. 

48. 3x + 2y, 2x + 3y, Ax + Ay. 

49. 5 x + 2 y, 3 x — y, 6 x + y. 

50. 3 x + 5 y, 7 x — 3 ?/, 5 ?/ + x. 

51. 7 c + 3 d, 2c+d, c — d. 

52. 4 c - 8 d, c+5d, c — 3d. 

53. oj + 9 y, x — 6y, x — y. 

54. # + 18, x — 15, Ay — x. 

55. c + 8, c-5, c-2. 

56. Find the upper base, lower base, altitude, and area in 
examples 49-55, if x = 3, ?/ = 2, c = 4, d = — 5. How do you 
account for your negative areas ? 

57. If 10' be subtracted from the length of a rectangle, and 
the same amount is added to the breadth, the area will be 
increased by 100 square feet, but if 10' be subtracted from the 
breadth and 10' added to the length, the area will be diminished 



MULTIPLICATION OF POLYNOMIALS 41 

by 300 square feet. Make a diagram of each rectangle, using 
a scale of y 1 ^ n = 1 foot. 

Division of Polynomials 

40. Polynomial by Monomial. Eeview division of mono- 
mials, §§ 32-35. Give sign rule for division. Give exponent 
rule for division. Give rule for subtraction of monomials. 

41. Since a polynomial is a sum of monomial terms, to 
divide a polynomial by a monomial, divide each term of the 
polynomial by the monomial and add the quotients thus found. 

Ex. 1. Divide 81 a 4 - 27 a 2 + 18 a by 9 a. 

81 a 4 -- 9a = 9a 3 , - 27 a 2 -f- 9 a = - 3 a, 18 a ■*- 9 a = 2. 
Then (81 a 4 - 27 a 2 + 18 a) -4- 9 a = 9 a 3 - 3 a + 2. 
The work should be written as follows : 

9a )81a*-27a 2 + 18q 
9 a 3 - 3 a + 2 

Ex. 2. Divide 21 a 4 - 18 ar* + 5 ar>-9 a by -3#. 

-3a Q 21x 4 -18x 3 + 5x 2 -9x 

- 7 x 3 + 6 x 2 - J z +3 

Ex. 3. Divide 4 a 2 (2 m+3) -9 o(2 m+3) +2 m+3 by 2 m+3. 

2?n + 3 )4 a 2 (2 m + 3)- 9 a(2 wi + 3) + 2 ro + 3 
4 a 2 -9a " +1 

Note that in dividing one number by another, we are simply 
removing from the dividend the factors found in the divisor. 

Ex.4. 45 -=-3. 

45 = 3 2 • 5. 

3 )3 2 -5 
3 -5 

After removing the 3 of the divisor 3 • 5 are left for the quotient. 



42 MATHEMATICS 

Ex. 5. 45 afy -s- 9 x\ 

45 x?y = 3 2 • 5 • x • x • x • ?/. 

9 x 2 = 3 2 • x • x. 
3 2 • x • x )3 2 ■ 5 ■ x ■ x • x • y 
5 . x • y = 5 xy 

If one carries this principle in mind no rule for division by a 
monomial is necessary. 

EXERCISE 17 

1. 72 x 7 y -5-24 afy. 

2. 4(a + 6) 5 — 4(a + &) 3 . 

3. (128 a 6 - 80 a 4 + 32)- 16. 

4. 128 a 6 -(80 a 4 + 32) -16. 

5. 128 a 6 - 80 x* + 32 -16. 

6. (128 x 6 - 80 a 4 ) + 32 -16. 

7. [7a 4 (2a + ^) 3 -12^(2a + ^) 2 +15a(2a + 6)]--(2a+6). 

8. Divide 14 a(x — y) + 7 a 2 (as — ?/) 2 — 49 cr (a — ?/) 3 by 

-la(x-y). 

9. Divide (a + b)a 2 + (a + 5)2 afr + b\a + 6) by a + b. 

10. Divide 144 afy 2 z - 729 x 2 ^ + 162 xyV by - 3 a; 2 ?/ 2 *. 

42. Polynomial by polynomial. 

The product of x 2 + # + 2 by 2 a + 3 is found by 

x\2 x + 3)+ a?(2 a + 3) + 2(2 a + 3) (1) 

= 2 ar 3 + 3 or + 2 a; 2 + 3 x + 4 x + 6 (2) 

= 2*r 3 + 5a; 2 + 7a + 6. (3) 

Ex. 1. Required to divide 2 x* + 5 x 2 + 7 a + 6 by 2 a + 3. 

This means that 2 x 3 + 5 x 2 + 7 x + G is the product of two factors or 
sets of factors, one of which is 2 x + 3. The problem is to find the other 
factor. 

If 2 x 3 + 5 x 2 + 7 x + 6 is written in form (1), division can be performed 
as in example 3, § 41, but if the multiplication has been completed and 
the partial products added as in form (3), the factor required is not so 
readily seen. 



MULTIPLICATION OF POLYNOMIALS 43 

It is evident from (1) that (3) is made up of partial products. If 
2 x + 3 is the divisor, x 2 + x + 2 is the quotient. 

An examination of (1) shows that the first term of each partial product 
is the product of the first term of the divisor by the corresponding term of 
the quotient. Therefore we may form this rule : 

1. Arrange both dividend and divisor according to the as- 
cending or descending powers of the same letter. 

2. Divide the first term of the dividend by the first term oi 
the divisor. 

3. Multiply each term of the divisor by the quotient found 
in 2. 

4. Subtract the product found in 3 from the dividend, 
arranging the difference found in the same order as the 
dividend. 

5. Divide the first term of the difference by the first term 
of the divisor. This gives a second term of the quotient. 

6. Proceed in this manner, considering each difference as 
a new dividend until the first term of the difference is of 
lower degree than the first term of the divisor. 

7. If there is a remainder, make it the numerator of a frac- 
tion whose denominator is the divisor, and annex with proper 

sign to the quotient. 

Dividend Divisor 



1st partial product, 


x 2 (2x + 3) = 


2d partial product, 


x(2 x + 3) = 


3d partial product, 


2(2 x + 3) = 


Ex. 2. Divide 


x s + 3x-2 by 




x s + 3 x - 2 




x s — 4 x 2 



2 x 3 + 3 x 2 



2 x + 3 



x 2 + x + 2 



2 x 2 + 7 x + 6 Quotient 
2 x 2 + 3 x 

4x + 6 

4x + 6 

4. 
x — 4 



x 2 + 4 z + 19 



74 



4 a* + 3 a - 2 * - 4 

4 x 2 - 16 x 

19x-2 
19 x — 76 
74 



44 MATHEMATICS 

EXERCISE 18 
Verify each result : 

1. Divide x\2 x + 3) + 8 x(2 x + 3) + 15(2 x + 3) by 2 x+3. 

2. Divide a\a + b) + 2 ab(a + b) + (a + 6) 2 by a + 6. 

3. The area of a rectangle is x 2 + 8 x + 15. The length is 
a; + 5. What is the breadth ? What is the breadth if x = 2 ? 

4. Divide ?/ 2 - 2 2/ — 15 by y + 3. 

5. Divide ?/ 2 + 2 y — 15 by y + 3. 

6. Divide y 2 + 2 ?/ — 15 by y - 3. 

7. Divide # 2 — 8 ?/ + 15 by y — 3. 

8. (5^-20a 2 + 15a-30)-v--5. 

9. 5^-(20a 2 + 15a-30)---5. 
10. 5a 3 -20a 2 +(15a-30)---5. 

11. 5y-20z 2 + 15a-30-5 + 3(2a;-8)-r-2. 

12. 4(a + &) 2 -J- 2(a + b) + 4(a + b) 2 + - 2 (a + b). 

13. The radius of a circle is ft + 3. Find the area. 

14. The area of a trapezoid is 2 x 2 + 12 a; + 18 ; the sum of 
the bases is 4 # + 12. Find the altitude. If the upper base 
is x + 7, what is the lower base ? If x = 3, what are the 
dimensions? Can you draw the trapezoid if it is isosceles? 

15. Divide ^4-3^ + 3z + lbya; 2 + 2a;-|-l. 

16. Divide a? + 3 x 2 + 3x + lby x + 1. Divide the quotient 
by x + 1. Compare results in examples 15 and 16. 

17. Divide x s + 27 by x + 3. 18. Divide a 3 + 1 by a + 1. 
19. Divide 8 a 3 - 27 b s by 2 a -3 6. 

20. Divide or 5 — 32 by 8 - 2. 

21. Divide a 4 — 81 by x — 3. 

22. Divide x A + 81 by x + 3. 



MULTIPLICATION OF POLYNOMIALS 45 

23. Divide x 2 — x — 12 by x — 9. 

24. The area of a trapezoid is 15 x 2 — 34 x + 16 ; the altitude 
is 5x — 8 ; one base is 2 a? + 3. Find the other base. 

25. The area of an isosceles triangle is 12 x 2 + 32 # -— 35 ; 
the base is 6x — 5. Find the altitude, then construct the 
triangle when x = 2. Construct the triangle when x = — 2. 

26. The. area of a rectangle is 4# 2 + 12&+9; the base is 

2 # + 3. Find the altitude. Construct the rectangle. Com- 
pare the base and altitude of your drawing. What kind of a 
rectangle is it? 

27. Divide the sum of x 3 + 7 x 2 + 35 x -f 19 and 

# 3 + 8 x 2 — 13 a; — 34 by the difference between 5x 2 -\-2 x—7 and 

3 x 2 _3 a _4 

28. Perform the following operation: (3 a + 7)(> 2 - a; - 12) ^ 

# — 4 

29. a 2 (2 a 3 + 9 x 2 - 71 a - 120) -s- (a? 2 + 3 x - 40). 

30. oj 2 (2^ + 9^-71^-120)---^ + 3^-40. 

31. Divide a 2 + 2 a& + 6 2 by a + b. Is a 2 + 2 a& + 6 2 a square 
or a rectangle ? Define a square. 

32. (100 x* - 229 a 2 + 9) h- (5 a - 1). 

33. Divide 100 x* - 229 x 2 + 9 by 4 a 2 - 9. 

34. Divide 100 a 4 - 229 x 2 + 9 by 25 a 2 - 1. 

35. Divide 100 x A - 229 x 2 + 9 by 5 x + 1. 

36. Divide 100 x* -229 x 2 + 9 by 2 x + 3. 

37. Divide 100 x* -229 x 2 + 9hy 2 x- 3. 

38. What are the factors of 100 x* - 229 x 2 + 9 ? of 4 a 2 - 9 ? 
of 25 x 2 - 1 ? 

39. Divide X s + 6x 2 - x-30 by x- 2. 

40. Divide a? 3 + 6 # 2 — a? — 30 by x + 3. 

41. Divide x s + 6 x 2 - x - 30 by x + 5. 



46 MATHEMATICS 

42. What are the factors of ar + 6 as 2 — x — 30 ? What does 
the product of these factors equal ? 

43. The length, breadth, and thickness of a rectangular 
solid are x-\-b, x + 3, and as — 2, respectively. Find its vol- 
ume. What are its dimensions when as = 3 ? x = 2? x = 1 ? 
# = 0? x= -1? 

44. Divide a; 2 — 64 by x — 8. 

45. Divide x 2 — 64 by x + 8. 

46. What are the factors of x 2 — 64 ? 

47. The volume of a circular cylinder is 

(16 x 3 - 84 x 2 + 120as- 25)tt. The radius of the base is 2 a? -5. 
Find the altitude. Find the dimensions when x = 3; 4; 5. 

43. The various symbols of algebra enable us to express 
many operations by using these symbols in place of words. 
In other words, the symbols are the shorthand, the stenography, 
of mathematics. 

For example : If we wish to indicate the subtraction of 
2 x — 1 from 4 x + 7, we may write : 

(a) from 4 x -f- 7 take 2 as — 1 ; or we may write : 
(6) (4as + 7)-(2as-l); 

the two expressions being identical, and read in the same way. 
In the next exercise be sure you translate the algebraic lan- 
guage into the English before attempting to solve the problems. 

REVIEW 

1. (2as + 3)(2as-7) + (3as-l)(2as+5). 

2. (2aj + 3)(3as + 2)-(4as + 15). 

3. (4as-l)(as + 4)-(as-4)(4as + l). 

4. as 3 - [3 as 2 + (3 as - 1)] - [a 8 + 3 x 2 - (3 x - 1)]. 

5. 4(as-5)(as-2)-4. 



MULTIPLICATION OF POLYNOMIALS 47 

6. 7 + 3-3-4. 

7. 2+(8-s-4) -5-8. 

a 6 -90 -5- (3. 10) + 2 -8. * 

9. (2x — T)x — (3 x + 4)a. 

10. Two right triangles are equal if two legs of the one are 
equal respectively to two legs of the other. Prove. 

11. In a rectangle the opposite sides are equal. Prove that 
the diagonal divides a rectangle into two equal triangles. 

12. The bisectors of the base angles of an isosceles triangle 
are equal. Prove. 

13. [(a + bf + 6 b(a + 6) 4 - 12(a + bf] -5- (a + 6) 2 . 

14. (4a 2 -^ 2 )-(2a + 6)(2a-6). 

15. (64c 2 -25c? 2 )-(8c + 5fZ)(8c-5cT). 

16. (5* + 2y)»-(5*-2y)(5»--2y). 

17. (5a; + 2?/)(5^-27/)-(25a; 2 -42/ 2 ). 

18. (7£-3z) 2 -(7a + 3z) 2 . 

19. (x + 5) (a? - 2)-(a> - 5)(a? + 2)-(a? - 5) (a? - 2) 

-(» + 5)(a + 2> 

20. (a + 5) 3 . 

21. (c + 7) 3 . 

22. (a 2 + a& + 6 2 )(a 2 - a£ + 6 2 ). 

23. (a 2 + ab + & 2 )(a 2 + a& + 6 2 ). 

24. Find the area of a trapezoid whose upper base, low T er 
base, and altitude are x + 7, 2 # + 8, 3^ + 2 respectively. If 
x = 1 and the trapezoid is isosceles, construct the trapezoid. 

25. Find the prime factors of (25) 2 ? (24) 2 , (72) 2 . 

26. What are the prime factors of (12) 6 ? (144) 5 ? 

27. What are the prime factors of 1728? of (39) 3 ? 

28. Factor (81) 4 , (8) 5 , (16) 10 . 



48 MATHEMATICS 

29. Factor 27 • 81 ■ 729. 

30. Factor 72 . 64 - 36. 

31. Factor 45 z s y 2 . How many prime factors in this num- 
ber? 

32. Factor 81 (a + b) 5 . 

33. Factor 27 (2 x — y) s . If x = 3 and y = 1, what are the 
prime factors of this product? How do they differ from the 
prime factors of (15) 3 ? 

Supplemental Applied Mathematics 

1. One egg weighs 2 ounces. 57% of the egg is white, 
32 % yoke, 11 % shell. Find the weights of the whites, yolks, 
and shells of one dozen eggs. 

2. One egg weighs 2 ounces. The shells of 2 eggs weigh 
one ounce. What per cent of the whole egg is the edible 
portion ? 

3. The edible portion of 2 eggs measures f of a cup. It 
takes 9 whites of eggs to measure 1 cup. How many yolks of 
eggs does it take to make 1 cup ? 

4. Beaten whites of 4 eggs measure 2-J- cups. What is the 
per cent of increase in quantity of beaten over unbeaten whites ? 

5. Beaten yolks of 3 eggs measure ^ cup. What is the per 
cent of increase in quantity of beaten over unbeaten yolks ? 

6. One egg, yolk and white beaten together, measures 4 
tablespoonfuls. How much greater is the increase in quantity 
when yolks and whites of 4 eggs are beaten separately ? 

7. According to Hutchison, experiments as to the difference 
in time of digesting eggs cooked in various ways show that 2 
soft "boiled" eggs leave the stomach in If hours, and 2 hard 
"boiled" eggs leave the stomach in 3 hours. If 2 eggs are 
eaten on each of 4 days a week, how many more hours' work a 
month will the stomach have in digesting hard than soft 
"boiled" eggs ? 



MULTIPLICATION OF POLYNOMIALS 49 

8. An egg was boiled for 3 minutes. After artificially 
digesting for 5 hours in a pepsin solution, it contained 8.3 % 
undigested protein. An egg was cooked in water at 180° F. for 

5 minutes. It was entirely digested after 5 hours in a pepsin 
solution. The edible portion of eggs contains 13.4 % protein. 
Find the weight of undigested protein from 1 dozen eggs if they 
are soft "boiled" rather than soft "cooked." 

9. Steel is 7.83 times as heavy as water. Find the weight 
of one cubic inch of steel. 

10. A gas range has four burners, each of which burns .65 
cubic feet per minute and two oven burners, each burning 1.032 
cubic feet per minute. Find the cost per hour of running the 
stove when all burners are on full, gas at 65 $ per thousand. 

11. A house is heated by a gas furnace containing four 
burners and two pilot lights. Each burner consumes one cubic 
foot in two minutes ; each pilot one cubic foot in eight minutes. 
Find gas bill for February at 30^ per thousand cubic feet. 
The pilots burned all the time. Two burners burned from 

6 a.m. to 10 p.m. and four burners were running two hours in 
the morning and two hours in the evening each day. 

12. According to one authority, a family of five living on 
$2000 to $4000 per year should spend 25% of that sum for 
food ; 20 % for rent ; 15 % for operating expenses, such as fuel, 
wages, etc., 15% for clothing; and 25 % for higher life, i.e. 
books, travel, charity, saving, and insurance. What amount 
should be used for each item if a family has an annual income 
of $3000? 

13. A family living on $1000 to $2000 per year requires 
25 % for food ; 20 % for rent ; 15 % for operating expenses ; 
20 % for clothing ; and 20 % for higher life. If a family lives 
on $ 1500 per year, what amount should be spent for each item ? 

14. From a family income of $800 to $1000 per year, 30% 
should be spent for food; 20% for rent; 10% for operating 
expenses; 10% for clothing; and 20% for higher life. 



50 MATHEMATICS 

How much can be spent for each item when the income is 
$900? 

15. A family living on $ 500 to $ 800 per year should spend 
45 % for food ; 15 % for rent ; 10 % for operating expenses ; 
10 % for clothing, and 10 % for higher life. If a family's 
annual income is $ 650, how much should be spent for each 
item ? 

16. From an annual income of less than $ 500, 60 % should 
be spent for food ; 15 % for rent ; 5 °J for operating expenses ; 
10 °/ for clothing ; and 10 % for higher life. If the income of 
a family is $ 425 per year, how much can be spent for each item 
per year and month ? 

17. The grocery bill of a family living on $ 3000 per year 
should be 25 % of that sum ; of a family living on $ 700 a 
year, 45 % of that sum. Find the difference in the monthly 
grocery bill of each ? 

18. 20 % of a twelve-hundred-dollar income should be spent 
for clothing, and 15 % of a nine-hundred-dollar income. Find 
the difference in the amounts spent for these items by two 
families having these incomes. 

19. There are 49 lb. flour in a fourth barrel, or an ordinary 
sack of flour. 1 lb. flour measures 4 cups. If a family uses 5 
loaves bread per week, and it takes 3J- cups of flour to make 
one loaf, how many months will a sack of flour last ? 

20. Find the cost of 4 loaves of bread, requiring 1 hour for 
baking and containing the following: 3^- qt. flour, 1 yeast cake, 
2 tb. lard, 4 t. salt, 4 t. sugar. Flour costs $ 2.00 per one-fourth 
barrel; yeast 2 per cake; lard 15^ per pound (2 c. in a 
pound), the sugar and salt $.0024; gas 70^ per 1000 ft., the 
oven burner burning 39 cu. ft. per hour. 

(Tb. = tablespoon, t. = teaspoon, c. = cup.) 

21. I buy 28-inch material for handkerchiefs. How many 
yards would I have to buy to make 9 dozen handkerchiefs, cut 
14 inches square ? 



MULTIPLICATION OF POLYNOMIALS 51 

22. What is the size of the finished handkerchief if a i-inch 
hem is made on four sides ? 

23. If I paid 2 cents a yard for stitching, how much would 
the work on 6 dozen handkerchiefs cost ? 

24. How much lace would it take to sew around 9 dozen 
handkerchiefs if you allowed 4 inches extra on each handker- 
chief for fullness ? 

25. If the material cost 60 cents a yard, and lace 15 cents a 
yard, how much would the handkerchiefs in problem 24 cost, 
including 5 spools thread at 5 cents a spool and 2 cents a yard 
for stitching ? 

REVIEW 

Divide : 

1. 2 a? - 5 x 2 + 8 x - 3 by 2 x - 1. 

2. x 3 — 4 cc 2 + 5 a; — 2 by x — 1. 

3. 2 a 3 - 9 a 2 + 8 a + 3 by a - 3. 

4. b z - 3 & 2 c + 3 5c 2 - c 3 by b - c. 

5. c 3 + 2 c 2 + 2 c + 1 by c + 1. 

6. x 3 + 2 # 2 i/ + 2 &?/ 2 + ?/ 3 by x + ?/. 

7. a 2 + 4 a - 21 by a - 3. 

8. a 2 + 4 a — 21 by a — 7. 

9. a 2 + 4 a - 21 by a + 3. 

10. a 2 + 4 a — 21 by a + 7. 

11. a 3 + 3 a 2 + 3 x + 1 by a 2 + 2 a? + 1. 

12. x 5 + 3 # 2 + x — 1 by x 2 + 2 a + 1. 

13. a 4 + 6 a 5 + 35 a 2 - 8 - 5 a 3 + 19 a by 3 a - 1 + 2 a 2 . 

14. # 4 — 2 ^2/ — ?/ 4 + 2 #?/ 2 by # 2 — y\ 

15. # 4 — 2 # 2 2/ 2 + ?/ 4 by x 2 — 2 xy + y 2 > 

16. 6 2/ 5 - 4 ?/ 4 + 20 2/ 3 - 13 y 2 + 6 i/ - 3 by ?/ 2 + 3. 

17. 6 m 4 + 13 m z — 70 m 2 + 71 m + 40 by 3 m 2 — 7 m + 4. 

18. z 6 - 2 z 5 + z 4 + z 3 - z 2 - 2 2; - 1 by z 2 - z - 1. 

19. z 6 - 7 a 4 + 6 x z - 35 a 2 - 60 x - 50 by x 2 - 10. 



52 MATHEMATICS 

20. [(a? - 1)(2 a? + 3)(4 a; 2 - 9)] by 2 a; 2 + x - 3. 

21. [(3 x - 1)(3 a> + 1)(4 x + 5)] by 12 a; 2 +11 a? - 5. 

22. [(125 x 3 - l)(aj + 7)(2 a? + 1)] by 10 x* + 3 x - 1. 

23. [(a 2 - l)(a 2 - 2 a - 15)] by a 2 + 2 a - 3. 

24. [( a + l)(2a + 3)(4a 2 -9)] by 2a 2 -a-3. 

25. [(2 x - 3)(5 a+ 7)(4 a- 3)(3 a?- 7)] by 6 a; 2 - 23 a; + 21. 

26. [(18a; 2 + 3a;-l)(10a; 2 + 13a;--3)] by 6 a; 2 + ll x + 3. 

27. 2a; 4 + 3a; 3 -5a; 2 -8a;-9by 2a; 2 + 7-5a;. 

28. a^ + y z + z 3 — 3 xyz hy x -\- y + z. 

29. 2 a; 4 + a; 3 - 6 a? 2 - 4 x - 8 by x — 2. 

30. 125 a; 6 - 512 by 5 a; 2 - 8. 

31. b A — 4 y 4 by 6 — y. 

32. 52 ar 5 + 38 a; 4 ?/ - 208 afy 2 - 143 xhf + 91 a;?/ 4 + 35 y 4 
by 2 a; 2 - 7 y 2 . 

33. l-a 6 by 1 - 2 a - 2 a 2 - a 3 . 

34. 6a; 4 — 13x?y + 13 a; 2 ?/ 2 - 13 a*/ 3 - 5 ?/ 4 by 3 a; 2 - 2 a;?/ + 5 # 2 . 

35. 2ar> - 3a; 4 - lSar* + 14a: 2 - 2x - 12 
by 2 a? 4 + 3 a; 3 - 4a; 2 + 2a; + 4. 

36. 16a; 4 + 36a; 2 + 81 by 4a; 2 - 6a; + 9. 

37. 10m 6 + 11 m 5 - 53 m 4 - 30 m 3 + 1 m 2 + 97m - 14 
by 2 m 2 + 3 m — 7. 

38. 12a; 5 - 11a? - 6a; 2 - 15a; + 10 by 3a; 2 - 5. 

39. 10 y 5 - 9y 4 - 13 ?/ 3 + Sy 2 - y - 3 by 2y 2 - y - 3. 

40. 21 a; 5 - 6x 4 - 22a; 3 + 26a; 2 + 5x - 8 by 3 x 2 - 1. 

41. A man died leaving an estate, worth 36 a? 4 — 109 a; 2 + 25 
dollars, to his 6 x 2 + 13 x + 5 heirs. If they receive equal 
amounts, find each one's share. 

42. A train went at the rate of 5 a; 2 + 32 x — 21 miles per 
hour for a; 2 + 6 x — 7 hours. How far did it go in 2 a; 2 — 5 x -f 3 
hours ? 

43. The area of a rectangle is 2 x 4 + 15 x? -\- 4 x 2 — 37 x — 24 
square rods. If the altitude is 2 a; + 3 rods, find the base. 



MULTIPLICATION OF POLYNOMIALS 53 

44. A and B both worked the same number of days. A 
received 3a? 4 — 2 # 3 — 5x 2 + 6x — 5 per day while B received 
2 x* — 2 x z + 7 x 2 — 9 x + 4 per day. At the end of the time 
they both received 10 x* — 13 a 5 + 3x* — 4ar* — x 2 + 4 a; + 1. 
How many days did they work ? 

45. The area of a triangle is 20 ar* — 2 x 4 — 4 a^ — 40 x 2 — 46 # 

— 12 square feet. If the base is 4 # 2 + 2 a; + 6 feet, find 
altitude. 

46. x 2 — 5 x + 6 teachers receive together 2 a; 6 — 19 x 5 + 50 x A 

— 13 a? 3 — 73 # 2 + 41 a; — 6 dollars. What is their average 
salary ? What is their average salary if a; is — 2 ? How 
many teachers if a? is — 2 ? 

47. [(2 a; 6 - 14 x 5 + 13 a* 4 + 33 x? - 17 x 2 - 23x - 4)- 

<y - 6 a; 4 + 5 x* -f 49 a; 2 - 98 x + 23)]-*-(a? - 6 a; 2 + 3 a; + 4)= ? 

48. Divide the sum of 21 x 2 + 20 xy — 7 x - 30 y 2 — 1 and 
9 a; 2 - 3 a;?/ + 17y — 5 ?/ 2 — 1 by 6 a; - 5 2/ + 1. 

49. Divide the difference between 9 a; 3 — 36 x 2 + 21 a; — 45 
and 3 a? 3 — 57 x 2 + 55 x by the difference between 4 x 2 — 7 a? + 12 
and 3 a; 2 + 12 x - 7. 

50. Divide the sum of 3 ar 3 + 3 x 2 - 6 x - 14 and 2 a? - 7 a; 2 
+ 9 cc -f- 11 by the sum of 3 x + 4 and 5 a; 2 — 2 a?. 

51. [(7 a 3 - 32 a 2 - 13 a + 29) + (8 a 3 + 6 a 2 + 6 a - 9)] 

-r- [(7 a 2 + 13 a + 1) - (2 a 2 + 15 a + 6)]. 

52. [(21 a s x + 4 a + 27 aa 2 ) + (4 a? - 21 a?x* - 27 ax)] 

-^ [(15 a 2 a? 2 - 20 ax + 3) + (6 a 2 a; 2 - 7 as + 4)]. 

53. The area of a rectangle is a 6 -f 3 a 4 — 6 a 2 — 18. Its 
altitude is a 2 + 3. Find its base. 

54. The area of a triangle is 12 a 4 — 12 a 3 + 8 a 2 — 4 a — 4. 
Its base is 6 a 3 + 4 a + 2. Find its altitude. What are its 
dimensions when a = 5. 

55. The area of a triangle is 2 a 4 — a 3 — 6 a 2 + 7 a — 2. Its 
altitude is 2 a 2 + 3 a — 2. Find its base. What are its dimen- 
sions when a = 4 ? 



CHAPTEE V 
Inequalities. Simultaneous Equations 

44. The sign of inequality is > or <, the opening being 
toward the greater number. 

Thus 9 > 5 is read, " 9 is greater than 5." 
4 < 7 is read, " 4 is less than 7." 
— 5 > — 9 is read, " — 5 is greater than — 9." 

45. Two inequalities are in the same sense when the first 
member of each is greater or less, respectively, than the second 
member. 

46. If equal numbers be added to or subtracted from both 
sides of an inequality, the resulting inequality is said to con- 
tinue in the same sense. That is, the inequality sign is not 
reversed. 

Ex. 9 > 5. Subtract 3 from each side. Then 

9_3>5_3 

or 6 > 2. 

47. If all of the signs of an inequality are changed, the in- 
equality does not continue in the same sense. 

That is, if all the signs of inequality are changed, the sign 
of inequality must be reversed. 

48. An inequality continues in the same sense after being 
multiplied or divided by a positive number. 

Ex. 1. 2 x > 10. 

Dividing both sides by 2, we have 

x>5. 
54 



INEQUALITIES 55 

Ex.2. 2x+y>8, (1) 

y = 2. (2) 

Subtract (2) from (1), 2 x > 6 (3) (§ 46) 

Divide (3) by 2, x > 3 (4) {§ 48) 

That is, to satisfy the above conditions, x must be greater than 3. 

Transposition 

49. In § 24 we solved an equation, 

2x + 3 = 9 

by subtracting 3 from each side, by Axiom 2, then dividing 
both sides of the resulting equation by the coefficient of the 
unknown number. 

By a method called transposition, it is possible to simplify 
the first operation. 

Ex. 1. Solve for x, mx + c = b. (1) 

Subtracting c from each side, mx — b — c. (2) 

Note that the equation (2) is the same as if we had removed -f c from 
the first member of the equation (1) and placed it in the second member 
with its sign changed. 

Such change is called transposition. 

This rule follows : A term may be transposed from one mem- 
ber of an equation to the other if its sign is changed. 

Then, in solving, mx -\- c = b. 
Transposing c, mx = b — c, 
and dividing both members of the equation by m, the coefficient of se, 

m 

Ex. 2. ax + 7 = 13 - 2 ax. (1) 

Transposing the unknown term — 2 ax to the first side, and -f 7 to the 
second side, we have ax + 2 ax = 13 — 7, (2) 

3 ax = 6, (3) 

6 2 ... 

x =r a =a (4) 



56 MATHEMATICS 

To verify this value of x, substitute the value of x in (1). 

a © +7=13 " 2a ©' 

or 2 + 7 = 13 - 4. 

9 = 9.- .'. is the abbreviation for hence. 

Simultaneous Equations 

50. In examples of exercise 8 and the last ones of exercise 10, 
we note that in the former exercise one unknown number is 
involved in the equation, while in the latter the values of two 
unknowns must be found. 

The method for the solution of equations in two unknowns 
is to find some way of combining the two equations into one 
equation containing one unknown, and solving this resulting 
equation. 

Such process is called elimination. 

Two equations in two unknowns are necessary because one 

equation in two unknowns has no definite solution, but has an 

indefinite number of solutions. 

E.g., in 
x + y = 6, if x = 0, y =6$ x = 1, y = 5 ; x = 2, y = 4; x = 3, y = 3; etc., 
any number of pairs of values satisfying the given equation. 

51. Note that in this equation y changes value when x 
changes. This is because x and y are so related that their sum 
must always be 6. Such values must be given x and y that 
the equation must be kept in balance; that is, must be kept an 
equation. 

The unknown numbers which are subject to change of value 
in an equation are often called variables. 

52. The usual methods of elimination are addition or sub- 
traction, substitution, and comparison. 

53. First Method. Addition or Subtraction. In this method 
one or both equations are multiplied in such a manner that the 



INEQUALITIES 57 

coefficients of the same unknown in both equations become 
identical. One unknown number may then be eliminated by 
addition or subtraction of the corresponding sides of the equa- 
tion. 



Ex. 1. 




2x + y = 9, 


(1) 






3x-2y = 10. 


(2) 


Multiply (1) by 2. 




4 x + 2 y = 18 


(3) 


Add (2) and (3), 




3 x - 2 y = 10 


(2) 


Whence, 




Ix =28 


(4) 


and 




x =4. 




Substitute x — 4 in 


(1), 


8 + y = 9. 


(5) 


Transposing (§ 49) 




+ 8 in (5), we have, 
y = 9 - 8, 




or, 




F = l. 





Check these values in (2), 

3.4-2-1 = 10, 

12 - 2 = 10. 

Ex.2. 8x- 5y = 31, (1) 

12® + 13#=-15. (2) 

Multiply (1) by 3, and (2) by 2, 

24 x -15 y = 93 (3) 

24 x + 26 y = - 30 (4) 

Subtract (4) from (3), - 41 y = 123. (5) 

Divide (5) by —41, the coefficient of y. 

Whence, y = -S. (6) 

Substitute value of y in (1), 

8s- 5(- 3) =31, 

8z + 15=31. (7) 

Transposing (§ 49) 8 x = 31 - 15. 

Sx = 16. (8) 

Whence, sc = 2. 

Check in (2), 12 • 2 + 13 (- 3) = - 15. 

24 - 39 = - 15. 

EXERCISE 19 
Solve the following equations and verify results : 
1. 3aj-4y = l, 2. 5^+7^ = 29, 

Ax + 3 y = 18. 5t~7u = l. 



58 MATHEMATICS 

3. 3v- 5z = 12, 7. \x + §y=-.±s-, 

8v+10z= -38. |a?-f y= -f . 

4. 7y + 8a?=— 31, 8. ia+i# = 6, 

12x-3y=-3. ix + ± y= 2i-. 

5. £x + 5y = 29, 9. 28 £- 45?/= - 17, 
6x + 7y = 4:l . 35 x- 27 y = 8. 

6. \x-\y=l, 10. 36a? + 25.y = ll, 

^ + i?/ = 5 . 24 x + 55 ?/ = - 31 . 

54. Second Method. Substitution. 

Ex. Solve 2x + y = 9, (1) 

3a- 2?/ = 10 . (2) 

Solve (1) for ?/, y = 9 - 2 x. (3) 

Show that this is transposition. 
Substitute the value of y in (3) in (2). 

3z-2(9-2x) = 10. (4) 

Then, 3x-18 + 4x = 10 

(How do you account for the signs of 18 and 4 x ?) 
Transposing and collecting, 7 x = 28. 

x = 4. (5) 

Substituting (5) in (3), y = 9 - 2 • 4, 

or, y = l. 

Check : 2-4 + 1=9. 

EXERCISE 20 

Solve by method of substitution (verify results) : 

1. 4 x 4- y = 6, 4. 4 J? - 6 8 = 3, 

3s + 4y = ll . 8fl + 3£ = l. 

2. 12 ?7i- 7v=-2, 
llm-12i; = -13 . 

3. 45-65=- 1, 
8JB + 3# = 3. 



5. 


15 y 


-8* = 


=n, 




35?/ 


+ 6 z = 


- S^ 


6. 


10*- 


— 8 u 






6* + 16u 


= -i. 



INEQUALITIES 59 

55. Third Method. Comparison. 

Solve eacli equation for the same variable and equate the 
values thus found. 

Ex.1. 2x + y = 9, (1) 

3x-2y=10 . (2) 

Solve (1) for y, y = 9 - 2 x. (3) 

Solve (2) for y, y= 3 x ~ 1Q . (4) 

A 

Equate the values of y found in (3) and (4) (Ax. 8). 

9-2x = 3 *- 10 - (5) 

2 

Multiply (5) by 2, 2^9 - 2 x = 3x ~ 10 Y 

(6) 



or, 


18-4x = 3x- 10. 


Solving (6), 


x = 4. 


Substitute in (3) , 


2/ = 9 — 8 




= 1. 


Check as before. 




Ex.2. 


8 x — 5 y = 31, 




12^ + 13?/ = -15. 


Solve (1) for x, 


%x = 5y + 31, 




b-^ + 31 

8 


Solve (2) for a:, 


12 x = -13?/- 15, 




^_ _13 */ -15 
12 


Equate the values of x in (3) and (4) (Ax. 8) . 




5 2/ _l_31__13^_l5 




8 12 



(1) 

(2) 



(3) 



(4) 



(5) 

Multiply (5) by 24, a multiple of the denominators (Ax. 3). 

Then, 15 y + 93 = - 26 y - 30. (6) 

(See § 53, and note that the terms of (6) are the same as those that 
compose equation (5), example 2, before they were combined into two 
terms.) 



60 MATHEMATICS 

Transposing the unknown term to the first member, the known to 
the second member, and solving, 

41 y = - 123, 

V= -3. (7) 

-15 + 31 



Substitute in (3), x = 

Check as before. 



8 

2. 



5. 


3 
6 x + 3 y = -, 

5 x — 3 y = 4. 


6. 


3a + 3# = 2, 
9a — 15?/ = —2. 



EXERCISE 21 

Solve by comparison, verifying each result: 

1. 3x+6y=7, 4. 15 x -11 y = 16, 
4a?-2y = l . 13 a + 11 y = 1. 

2. 5 2/ + 4 a? = 6, 
2y- ar = l. 

3. 8 x + 15 y = — 8, 
43y-12x = 12. 

56. In stating written problems it is often convenient to 
use two or more unknown numbers rather than form a single 
equation in one unknown as was done in exercise 8. Care 
must be taken to form as many equations as there are un- 
known numbers. 

Ex. A number is composed of two digits. One half the 
number is equal to twice the sum of the digits, and if 18 is 
added to the number the digits of the resulting number will be 
those of the original number written in reverse order. Find 
the number. 

Let t = the digit in tens' place, 

and u — the digit in units' place. 

Then, 10 t + u = the number. (Exercise 2, example 14.) 

By the conditions, 10g + M = 2 (t + w), (1) 

Jj 

10* + u + 18 = 10?* + t, 

or *)t — \)u=- 18, 

t- u=:-2. (2) 



INEQUALITIES . 61 

From (1) 6 t - 3 u = 0, 

or 2t- u = 0. (3) 

Subtracting (2) from (3) £ = 2. (4) 

Substituting the value of t found in (4) in (2) 

2 - u = - 2, 
and w = 4. 

Whence, 10 1 -f w = 24, the required number. 



EXERCISE 22 

1. Find two numbers such that the sum of 3 times the first 
and 5 times the second is 24, and the sum of 1 of the first and 
\ of the second is 2. 

2. The altitude of a trapezoid is 8 r and its area 48 square 
feet. If the lower base is increased 4', its area will be increased 
16 square feet. Find its upper and lower bases. (Any trouble 
here ?) 

3. A cubic foot of steel and a cubic foot of water together 
weigh 5521 pounds. The difference between their weights is 
4271 pounds. How many times heavier than water is steel ? 
This quotient is called the specific gravity of steel. (Remem- 
ber these results.) 

4. The perimeter of a rectangular field is 240 rods, and its 
length is 40 rods more than its breadth. Find its area. 

5. A field of wheat is 80 rods longer than it is wide. The 
farmer uses a combined harvester and thresher that cuts an 11- 
foot swath. After making 15 rounds, the indicator shows that 
271 acres have been cut and have yielded 660 bushels. Find 
the yield per acre and the dimensions of the field. 

6. Find two numbers such that n times the first added to k 
times the second is c, and k times the first minus n times the 
second is b. 

7. The difference between the bases of a trapezoid is 6, the 
area 120, and the sum of the bases 30. Find the dimensions. 
Can you construct the trapezoid ? 



2x — y + 5 z = 


= 15, 


3x+2y-3z = 


= -2; 


4.x — 2y + 7z = 


= 21. 



62 MATHEMATICS 

57. When a problem involves three variables the method of 
solution is similar to that in two variables. Care must be 
taken to eliminate the same unknown in each equation. 

In solving simultaneous equations it is necessary that there 
be as many equations as there are variables in the problem. 

Ex. 2x - y + 5 z = 15, (1) 

(2) 
(3) 

Multiply (1) by 2 and add the result to (2), 

7 x + 7 z = 28. (4) 

Add (2) to (3), 7 x + 4 z = 19. (5) 

We now have two equations in x and z. 
Subtract (5) from (4), 3 s = 9, 

z = 3. (6) 

Substitute value of z in (5), 

7 x + 12 = 19, 
or / x == 1. (7) 

•Substitute (7) and (6) in (1), 

2-2/ + 15 = 15, (1) 

or ?/ = 2. 

Check as before, substituting these values in (2) and (3). 



EXERCISE 


23 


Solve the following : 






1. 2x + 3y-4z = 12, 

x — 2y+5z= — 6, 


4. 


1 1,3 25 

-X y 4- -z = — , 

2 4 J 2 4' 


3x + 2y-6z = 13. 

2. 5y — 3x + 2z = 5, 
5z-3y + 2x=-9, 
~)x-3z + 2y = 12. 

3. 3x— 2y=o, 


5. 


1,2 3 n 

-# + -?/ 2 = 0, 

5 5 J 5 

2 2,1 5 

3 3* 3 3 


1,1,1 13 
4 3 l/ 2 12' 


3y-2z = 25, 

3z-2x=-25. 




1,1,1 13 

-x + -?/ + -z = — , 

3 2*' ^4 12' 

1 ,1 ,1 13 

-X + -V + -Z = 

2 4' 3 12 



INEQUALITIES 63 

6. An examination contained 15 problems. Each pupil 
received 7 credits for a problem solved, and 3 demerits for a 
problem missed. One boy had 45 marks more to his credit 
than he had against him. How many problems did he miss ? 

7. Find 3 numbers such that if 3 times the first be added to 
one half the sum of the second and third, the sum will be 137. 
If half the sum of the second and third be subtracted from the 
first, the difference will be 23, the sum of the three numbers 
being 74. 

8. If the order of the digits of a certain three-place number 
be inverted, the sum of this number and the original number is 
444, and their difference is 198. The first digit is equal to the 
sum of the other two. Find the number. 

9. A merchant buys lemons, some at 4 for 5 cents, others at 
3 for 4 cents, spending $ 2.80. Again he buys as many of the 
first kind as he had bought of the second kind, and as many of 
the second kind as he had bought of the first, this time spend- 
ing $2.78. How many of each kind were purchased? 

10. There are two terms such that the sum of seven times 
the first and fourteen times the second is 221 while the sum of 
three times the first and five times the second is 9. What are 
the terms ? 

11. One fifth of the length of a rectangular lot is 40' less 
than its width. Its perimeter is 320'. If willow trees, ten 
feet apart, are to be set out as a wind-break, how many will be 
needed on the longer side ? (The first tree is to be planted in 
the corner of the lot.) 

12. There are two numbers such that if the first be increased 
by 6 and the second diminished by 10, the numbers will be 
equal. But if the first be diminished by 10 and the second 
increased by 6, the first will then be ± the second. What are 
the numbers ? 

13. One boy raised 14 bushels of potatoes more per acre 
than another boy. The first boy had 10 acres and the second 



64 MATHEMATICS 

had 12 acres. Together they raised 1790 bushels. How many 
bushels per acre did each raise ? 

14. The perimeter of a triangle is 572'. Two sides are equal. 
The third side is 20' shorter than the sum of the other two. 
What is the length of each side ? 

15. The sum of 2 # and 4 y is 4 less than the sum of 5 a? and 
7 y. Half the sum of x and y is equal to 7 x. Find the values 
of x and y. 

16. There are two numbers such that the sum of -^ of the 
first and T 5 6 of the second is 12. The sum of i of the first and 
\ of the second is 24. What are the numbers ? 

17. A croquet ground is 18' longer than it is wide. If its 
width be multiplied by 3, it will then be 13' more than its 
length. What are the dimensions ? 

18. What are the dimensions of a basket ball floor whose 
perimeter is 245', if, when the floor is used for two games each 
half fails to be a square by 5' ? 

Solution of Inequalities 

58. In solving inequalities the rules for solving equations 
hold except in some instances where change of sign takes 
place (§§44-48). 

EXERCISE 24 
If x be positive, which fraction has the greater value ? 

1 ^+J? or *±I 2 2x + 9 or 2x + 5 

4 6* 62 

(Test 1 and 2 by substituting values for x.) 

Find the limits of x for which these are true, and show that 
your results are correct. 

3. 3x + 5<x + 2* 5. 6a-4<0. 

4. 5*-p'>2* + 7. 6. 7a? + 14 >0. 

* Solve as you would an equation. 



INEQUALITIES 65 

Find the limits of the values of. x which satisfy the following 
inequalities simultaneously (verify your results) : 

7. 2 x + 6 > 0. (1) 

-x + 5>0. (2) 

From (1), x>-3. (3) 

From (2), x- 5<0 (§47), 
and x < 5. 

Then, to satisfy the conditions of (1) and (2), x must lie between — 3 
and -f 5. 

8. 24-3a>0, 10. 4.x -7 <0, 
16+2x>0. 6x-2x>0. 

9. 5x + 10>0, 11. 5x + 2 >0, 
3a;- 9<0. 2j;-5^>0. 

What values of x and y satisfy the following (§ 48) : 

12. 2x + 3y>10, 
3x-2y = 8. 

13. Seven times the number of boys in the algebra class less 
45 is greater than five times their number added to 13, and 
six times the number of boys less 22 is less than four times 
the number of boys added to 40. How many boys in the 
class ? 

14. Find the digit such that twice the digit increased by 3 
is greater than one fifth the digit increased by 11. 

15. Find a multiple of 18 such that five times the number 
decreased by 220 is greater than three times the number de- 
creased by 42. Is there more than one such multiple ? 

16. Find a multiple of 16 such that one half of it decreased 
by 28 is less than one third of it increased by 16. Does more 
than one multiple satisfy this condition ? 

17. Find a line containing an integral number of inches 
such that five times the length of it decreased by 165 is 
greater than three times its length increased by 14. Is there 
more than one such line ? 



66 MATHEMATICS 

Supplemental Applied Mathematics 

1. The 20tli Century Limited of the Lake Shore Railroad 
runs 54.5 miles per hour. Find the speed in feet per second. 

2. There are 24 gas stoves in the domestic science kitchen. 
If each stove burns 2.32 cubic inches of gas per second, find 
the cost of gas for one class of pupils during a period of 80 
minutes. Allow 15 minutes waste time (when stoves are not 
in use). (Gas at 75^ per thousand cubic feet.) 

3. The power of running a motor in the pattern shop is .044 
Kilowatt per hour. During a period of 80 minutes, 24 motors 
are running on an average of 60 minutes. Find the amount of 
power used. Find the cost at $ 0.0635 per Kilowatt hour. 

4. The Southern Talc Company is able to ship freight in car- 
load lots, by the long ton, and charge its customers the cost price 
per ton F. O. B. North Carolina, and base its charges on the 
short ton. Find the Talc Company's profit on 10 cars of 34 
tons each, shipped from North Carolina to Chicago. Freight 
rate $3.50 per ton. 

5. To find the weight of timber in the rough : multiply 
length in feet by breadth in feet by thickness in inches and 
multiply that product by one of the following factors. For 
Oak, 4.04 ; Elm, 3.05 ; Yellow Pine, 3.44 ; White Pine, 2.97. 
The result is in pounds. Measure six pieces of lumber in the 
shops and find the weight of each piece. Make accurate draw- 
ings of each piece. 

6. The weight of a grindstone is found by multiplying the 
square of the diameter in inches by thickness in inches and 
this product by .06363. 

(Diameter) 2 (Thickness) (.06363) = weight in pounds. (Re- 
member this formula.) Find weight of grindstone in your 
shop. Make diagram. 

7. The weight of one cubic foot of water is 62 V pounds. 
Find the weight of one gallon. 



INEQUALITIES 67 

8. Stock must be ordered for making 10,000 steel pins, each 
<!" diameter by 2" long. The rods for making these pins are i" 
in diameter and 12-0" long. Allow T y waste in cutting each 
pin. How many rods will be needed ? What will they weigh ? 

9. For flavoring, 1 ounce of chocolate is equivalent to 2 table- 
spoonfuls of cocoa. Chocolate costs 22 cents a cake (8 ounces), 
while cocoa costs 25 cents per box (-J- pound). There are 2 cups 
of cocoa in one box. Which is cheaper to use ? 

10. Chocolate contains 48.7% fat, and cocoa, 28.9%. If 
cocoa were used in place of 2 ounces of chocolate, how many 
tablespoonfuls of butter would need to be added to make the 
same quantity of fat as in chocolate, butter containing 85 % 
fat, and one tablespoonf ul weighing \ ounce ? 

11. Butter contains 85% fat, cream 18.5% fat, milk 4% fat. 
Calculate the quantity of butter necessary to use with 2 cups 
of milk to produce the same quantity of fat as 2 tablespoon- 
fuls of butter and 2 cups of cream, one pint of cream weighing 
one pound, and one pint of milk one pound and one ounce. 

12. Potatoes pared and then boiled lose 2.7% starch; pota- 
toes boiled with the skins on lose .2% starch. Find difference 
in quantity of starch lost by cooking 6 potatoes, one potato 
weighing 4^ ounces. 

13. Boiled Irish potatoes contain 20.9 % carbohydrates. 
Cooked sweet potatoes contain 42% carbohydrates. An Irish 
potato weighs 4^ ounces, and a sweet potato 6 ounces. Find 
difference in quantity of carbohydrates in six of each kind of 
potatoes. 

14. Find the reciprocal of 5280. Would you rather divide a 
number by 5280 or multiply the same number by its reciprocal ? 
Which would give the larger result ? Why ? 

15. Multiply 18649 cubic inches by the reciprocal of 1728. 
Is the result cubic feet ? 

16. Find the weight of a grindstone 36" in diameter and 4^-' ( 
thick. What will it weigh after it has been worn down 2\" ? 



68 MATHEMATICS 

17. A class of 15 girls wish materials and thread for cooking 
aprons. Each apron requires 2\ yards muslin at 12^ per yard, 
and a spool of thread at 5 p. How much material and thread 
for the class ? 

18. How much money was spent in making the purchases in 
example 17 ? How much does each girl pay for her share ? 

19. If a girl earned 22\f a piece making these aprons, how 
many would she have to make to earn $ 5.50 a week ? 

20. If these aprons sold at 85^ each, what would be the 
profit per apron ? 



CHAPTEE VI 





Lines, Angles, Triangles 

59. A portion of space is called a geometric solid. The 

boundary between the solid and the remaining space is called 
the surface of the solid. Eon example, if the sides, bottom, and 
top of a box were considered to have 
no thickness, they might be called the 
surface of the solid. 

60. If two surfaces intersect, their 
intersection is called a line. Thus, if 
^ jy surfaces AB and DO intersect in XCO, their in- 
tersection, XCO, is called a line. 

61. An intersection of two lines is called a 
point. Thus, F, the intersection of lines AB 
and CD, is a point. 

62. A line may also be regarded as — ^ 

the path of a moving point. " """" "' 

63. The line is said to be straight when it does not change 
its direction at any point. 

A line has no limit in its length. When a part of it is in- 
dicated, we speak of it as a segment or sect of the line. Thus, 

A B, AB indicates a straight line passing 

through points A and B. The part of the line between A and 
B is a segment or sect of the line. A line may also be indi- 
cated by a small letter placed somewhere on the line. 

ff-g., a is read " the line a." 

69 




70 MATHEMATICS 

64. A curved line changes its direc- 
tion at every point. E.g., the curve 
O ABC 

65. A broken line is composed of segments of successive 
BE F s t ra ig n t lines which have ' different 

A yS \ V " ~~J directions and which, pair by pair, 

^ ^ have a point in common. E.g., 

ABCDEFG is a broken line. 

66. The word line, unmodified, usually means a straight 
line. 

67. A plane is a surface such that if any two of its points 

be joined by a straight line, the 
line will lie entirely in the sur- 
face. 

Thus, in plane MN, if any two 
points, A and B, are joined by a 




line, the line will lie entirely within the surface. 

68. Plane Geometry treats of combinations of points and 
lines all lying in the same plane. 

69. A circle is a portion of a plane bounded by a curved 
line every point of which is equidis- 
tant from a point within called the 
center. 

The curved line is the circumference. 

The distance from the center to any 
point in the circumference is the radius, 
as CD. 
An arc is any part of the circumference, as AB. 




LINES, ANGLES, TRIANGLES 71 

Constructions 

To draw a circle, select a point on your paper and mark it 
C. Open your dividers to the distance required for the radius 
of the circle and place the metal point in the point C. With 
the pencil draw a curved line around the point O. The curved 
line is the circumference of the circle and the point C is the 
center (Fig. 1). 




Fig. 1. Fig. 2. Fig. 3. 

1. Draw a circle having a radius of one inch. 

2. Draw a circle having a radius of two inches. Draw two 
additional circumferences from the same center (Fig. 2). 

Circles that have the same center are called concentric circles. 

3. Draw a straight line AB. With A as a center and one 
half of AB as a radius, draw a circle. With 5asa center and 
the same radius, draw a circle (Fig. 3). 

At how many points do these two circles cut each other ? 

To draw the diameter of a circle, draw a straight line 
through the circle and its center. Mark the points at which 
this line touches the circumference A and B, respectively. 
The line AB is the diameter of the circle (Fig. 4). 

The diameter divides the circumference into two equal arcs 
and the circle into two equal parts called semicircles. 

4. Draw the diameter of a circle having a radius of one and 
one-half inches. 

5. Draw the diameter of a circle having a radius of two 
inches. 



72 



MATHEMATICS 



To divide a circle into four equal parts, draw a circle and 
its diameter (Fig. 4). Then, with A as a center and a radius 
a little longer than one half the diameter, draw a short arc 
directly above the middle of the upper arc of the circle. 




Fig. 4. 




With B as a center and the same radius, draw another short 
arc crossing the first and mark the point of intersection 0. 
Place the edge of your ruler in line with the point and the 
center of the circle and draw another diameter of the circle. 
Mark this diameter ED (Fig. 5). 

The two diameters divide the circle into four equal sectors 
and the circumference into four equal arcs. Each sector and 
each arc is called a quadrant. The two diameters make four 
equal angles at the center, called right angles, and are, there- 
fore, said to be perpendicular to each other. They cut each 
other into two equal parts and are, therefore, said to bisect 
each other. 

6. Draw a circle having a radius of one inch and divide it 
into four equal parts. 

7. Draw a circle having a radius of two inches and divide 
it into two equal parts ; into four equal parts. 

To measure angles, arcs, and sectors the circumference of a 
circle is divided into 360 equal parts called degrees. How 
many degrees in a semicircle ? In a quadrant ? In a right 
angle ? 



LINES, ANGLES, TRIANGLES 



73 



To bisect a straight line, as the line AB, draw short arcs 
above and below the middle of the line, with A as a center 
and a radius a little longer than one half the line. Then, with 
B as a center and the same radius, draw short arcs crossing 
the first two. Mark the points of intersection C and D, re- 
spectively, and draw a straight line through these points. 
The line CD bisects the line AB. Mark the point where the 
two straight lines cut each other (Fig. 6). 



M/c 



/\ 



~ o 

x 

Fig. 6. 

8. Draw Fig. 6. Then select the proper center and draw the 
circumference of a circle to show that AB is the diameter of a 
circle. Show also that CD is a diameter of the circle and that 
the four angles at the point of intersection of AB and CD are 
at the center of the circle. Show that the four angles are 
right angles. 

9. Draw a straight line, AB, two inches long and bisect it. 
Select a point on this line as a center and draw a circle to 
show that the line AB is the diameter of a circle. Then, with 
A as a center and the same radius draw another circle. At 
how many points do these two circles cut each other ? Mark 
these points M and JV", respectively. Draw lines connecting 
the point M with the centers of the two circles. 

What kind of figure do these lines form with the line join- 
ing the centers of the circles ? 

Measure the relative lengths of the lines of the figure thus 
formed. Can you give a reason for your answer ? 



74 



MATHEMATICS 



To inscribe a square in a circle, draw the circumference of a 
circle. Draw two diameters, AB and CD, cutting each other 
at right angles (Fig. 5). Draw straight lines connecting A 
with D, D with B, B with 0, and with A. The figure 
ADBC is a square. The points of the square where the sides 
meet lie within the circumference of the circle. The square 
is, therefore, said to be inscribed in the circle (Fig. 7). 
c 



Ruler 





10. Draw a circle having a radius of one inch and inscribe a 
square in it. 

11. Draw a circle having a radius of two inches and inscribe 
a square in it. Bisect the sides of the square and draw di- 
ameters dividing the inscribed square into four small squares. 

It is often necessary to draw one line parallel to another. 
When two lines have the same direction they are said to be 
parallel. 

To draw a line parallel to a given line AB, Fig. 8, take any 
point C not on the line AB. Lay the longest side of your 
drawing triangle on the line AB. Lay a ruler against the 
shortest side of your triangle. Now slide your ruler and 
triangle along the line AB until the point C touches the ruler 
on the side next to your triangle. Hold your ruler fixed and 
slide your triangle along the ruler until the point touches 
the longest side of your triangle. Draw OS along this longest 
side. CS is the required parallel. 

12. Draw Fig. 7. Then draw a line parallel to any one of 
the four sides of the inscribed square. 



LINES, ANGLES, TRIANGLES 75 

< 

Problem 

70. To draw a straight line equal to a given straight line. 

A* 1 •B 

I 

Given a line MN. 

To draw a straight line equal to MN. 

1. Draw any line AB longer than MN. 

2. With ilasa center and MN as a radius, describe an arc 
cutting AB at D. 

3. AD is the required line. 

71. An angle is the figure formed by two lines drawn from a 
point. 

The point is called the vertex. 

The above angle is read ABC, the letter at the vertex always 
^ being between the other two. The angle may 

also be designated by a small letter placed 
between the sides of the angle, as the angle x. 
This may be written Z x* The size of an 
angle does not depend upon the length of the 
sides, but upon the difference in the direction of the sides. 

72. Two angles are said to be adjacent when they 
have a common vertex and a common side between 
them. 
Thus Z a and Z b are adjacent. 

73. Two angles are equal when one can be so applied to the 
other that they will coincide throughout. 

* Also when only one angle is formed at a vertex, it may be read by a single 
letter, as ZB, read " angle at B." Small letters denote values ; capital letters 
denote position only. 





76 



MATHEMATICS 



Problem 
74. Jo construct an angle equal to a given angle. 



Given /.ABC. 

To construct an angle equal to it. 

1. Draw any line MN. 

2. Take any point D on MN for 
the vertex of the angle. 

3. With B as center and any 
radius BL, describe an arc cutting AB 
and BC at L and K, respectively. 

4. With BL as radius and D as 
center, describe an arc cutting DN at 
0. 

5. With as center and distance 
LK as radius, describe an arc cutting 
the former arc at E. 

6. Draw DEF. 

7. Then NDF is the required angle. 




75. A theorem is a truth requiring proof. 
In proving theorems the following axioms, in addition to 
those in § 23, are useful : 

Ax. 9. But one straight line can be drawn between two points. 
Ax. 10. A straight line is the shortest distance between two 
points. 



76. A triangle is a portion of a plane bounded by three 
straight lines. The lines are called the sides of the triangle. 
The points where the sides meet are the vertices. 

Three-sided figures play an important part in the study of 
geometry and in its applications to many problems in mathe- 
matics. Much of the work in measurement, in surveying, and 
in other kinds of engineering depends upon the triangle. 



LINES, ANGLES, TRIANGLES 



77 



Theokem I 

77. Two triangles are equal when two sides and the included 
angle of one are equal, respectively, to two sides and the included 
angle of the other. 



Draw a A ABC* 

On any line DE take DN equal 
to AB (§ 70). 

At D construct ZD = AA 
(§ 74). 

On Intake DM = AC and draw 

mn. 

In the above theorem, we have 
three equations given. (This given J) 
part is called the hypothesis.) 




sF 



X' 



^ 



•^r 



The given equations are : 

AC = DM 

AB = DN 
ZA = ZD 



(Don't forget that tvhat is 
given, the parts that are given, 
are the tools you have to work 
with.) 



To prove the geometric equation, 

A ABC = A DNM. (This is called the conclusion.) 

Proof. 1. Place A DNM on A ABC in such a manner that 
DN will coincide with its equal AB, D falling on A. 

2. DM will take the direction AC because AD — A A. 

3. Point if will fall on point C since DM= AC 

4. Hence line JOT will coincide with line CB (Ax. 9). 

5. Then A ABC = A DNM, since they coincide throughout. 



* In reading the letters at the vertices of a triangle always read counter- 
clockwise, beginning as far as is practicable at the lower left-hand corner, e.g., 
ABC, DNM in above triangles. 




78 MATHEMATICS 

Theorem II 

78. Two triangles are equal when a side and two adjacent 
angles of the one are equal, respectively, to a side and two adjacent 
angles of the other. 



/ M\ 

D< H >JT 

What is the hypothesis ? What is the conclusion ? 

Draw a A ABC. 

Draw a line DN equal to AB. 

At D construct an angle equal to Z A. 

At N construct an angle equal to Z B. 

Extend the sides of these angles to meet at M. We have 

Given in the A ABC and DNM 

DN = AB, 
ZD = ZA, 
ZN=ZB. 

To prove A ABC ±= A DNM. 

Proof. 1. Place A DNM upon A ABC in such a way that 
DN will coincide with its equal AB, point D falling on A. 

2. Since Z D = Z A, side jMf will take the direction AC 
and ilf will fall somewhere on AC. 

3. Since ZN= Z B, side A^Jf will take the direction 12(7 and 
Jf will fall somewhere on _B(7. , 

4. Since 3f falls on both AC and 2?C, it must fall at C, their 
point of intersection. 

5. Hence A ABC and DNM coincide throughout and are 
equal. 



LINES, ANGLES, TRIANGLES 79 



Problem 
79 . To bisect a given angle. 



E 




1. Draw an angle DCE. 

2. With. C as center and any radius CM, describe an arc 
cutting CE and CD at M and N, respectively. 

3. With M and N as centers and a radius greater than one 
half MN, describe arcs intersecting at 0. 

4. Draw CO. 

5. Then CO is the bisector of Z DCE. 

80. The right angle. If one straight line meets another 
straight line in such a manner that the adjacent angles (§ 65) 
formed are equal, the angles are called right angles and the 
lines are perpendicular to each other. 

Thus, if AB is a straight line and Z x = Z y, x and y are 
each right angles, and MP is perpen- 
dicular to AB. p* 

(_L is the abbreviation for "perpen- . x\y -& 

dicular to.") M 

81. If one straight line meets another straight line, adjacent 
angles are formed. If these angles are not right angles, the 
one greater than a right angle is called obtuse. The one less 
than a right angle is called acute. Either angle is said to be 
oblique. 



80 



MATHEMATICS 



\K 



CC/ 




C 



vv 



D 



82. If a line CD meets another line AB, two adjacent angles 
x and y, are formed. Suppose y less than x. If CD is made 

to rotate toward A, about Das a 
pivot, A x diminishes and Ay in- 
creases. It is evident that at 
_£ some position DK of DC, Ax will 
equal Ay. There can be only 
one such position. Then, one and only one perpendicular can 
be erected to a given line at a given point in the line. 

83. It follows from § 82 that : 

(a) Two right angles are equal. 

(b) If one straight line m.eets another straight line, the sum oj 
the adjacent angles formed is equal to two right angles. That is, 
in the figure of § 82. 

A x + A y = A ADK + A KDB = 2 rt. A. 

(c) TJie sum of all the angles on the same side of a straight line 
at a given point is equal to two right angles. 

(d) The sum of all the angles about any 
point in a plane is equal to four right angles. 

a + b + c + d + e +/= 4 rt. A. 

(e) If the sum of two adjacent angles is 
two right angles, their exterior sides lie in 
the same straight line. 

Let a + b = 2 rt. A. 




Extend PM to E. If AM and MB 
were not in a straight line, the sum A~ 
of c and d would not be two right 
angles, and this is contrary to (d). 




9m d 



~B 



'E 



84. Species of triangles. 

The right triangle is a triangle one of whose angles is 90 c 



LINES, ANGLES, TRIANGLES 



81 



or a right angle. The side opposite the right angle is 
called the hypotenuse. The other two sides are called the 
legs. 

All triangles except right triangles are oblique triangles. 

If an oblique triangle contains one obtuse angle, it is called 
an obtuse angled triangle. 

An acute angled triangle has each angle less than a right 
angle. 

Triangles are also named according to their lengths of 
sides. 

A scalene triangle has no two sides equal. An isosceles 
triangle has two sides equal. An equilateral triangle has three 
sides equal. 



Theorem III 



85. In an isosceles triangle the angles opposite the equal sides 
are equal. 

Draw line AB. With A and B as 
centers and the same radius greater 
than i AB, describe arcs intersecting 
at P. "* Draw PA and PB. We now 
have the isosceles triangle ABP, with 
b = a. 

What is given ? 

To prove Z B = Z A. 

Proof. 1. Draw line PM bisecting 
APB (§ 79). 

2. In A A MP unci MBP, 

b=a\ h = h] Az = Ay. 

3. Hence, A AMP = A3IBP 

(§ 77). 

4. Then AB — AA, being like parts of equal figures. 




82 



MATHEMATICS 



86. In equal figures, corresponding lines or angles are called 
homologous. It follows that in equal figures homologous parts 
are equal. 

Note. In equal triangles equal sides lie opposite equal angles. 
Thus, in equal A AMP and MBP, § 85, side a = side b, then 
Z A is homologous to Z B. 

Problem 

87. To construct a triangle when three sides are given. 

Let the given sides be a, b, c. 

1. Draw an indefinite 
line AIL 

2. On AK take AB = c. 

3. With A as center and 
b as radius, describe an arc. ^ 

4. With B as center and c 
a as radius, describe an arc 
intersecting with the former arc at C. 

5. Draw AC and CB. 

6. Then ABC is the required triangle. 




Theorem IV 

88. Two triangles are equal when three sides of one are equal, 
respectively, to three sides of the other. 

Draw a triangle ABC. 

Construct a second triangle DEF whose sides are equal to 
x, b, c, respectively (§ 87). Namely, d =a, e = b,f=c. 

What is given ? 
To prove A DEF = A ABC. 

Proof. 1. Apply A DEF to A ABC in such a way that / 
will coincide with its equal c, and F will fall opposite C 
2. Draw CF. 



LINES, ANGLES, TRIANGLES 83 

3. In A AFC, b = e. (By Hypothesis.) 

4. Then Zx = Zy (§ 85). 

(In an isosceles triangle the angles opposite the equal sides are equal.) 




t 



* s 

e/ \d 



d* - '* 

5. Similarly, AFBC is isosceles and Zn=Zm. 

6. Adding the equations in steps 4 and 5, 

Zx + Zn = Zy + Zm. 

7. Then, in A ABC and AFB, 

b = e 
a=f 

Z ACB, or Z x + Z n = Z .4i^B orZ?/ + Zm. 

8. Hence, A ^40B = A AFB (§ 77). 

(Two triangles are equal when two sides and the included angle of the 
one are equal, respectively, to two sides and the included angle of the 
other.) 

9. But A AFB is the same as A DEF. 
10. Hence, A DEF= A ABC 

Note. The base of a triangle is the side on which it is sup- 
posed to rest. Any side may be considered the base. The 
altitude is a perpendicular (§ 80) from any vertex to the oppo- 
site side. Theorems are truths to be used. Bemember each 
one. Notice that theorems I and III are both used in proving 
theorem IV. 



84 



MATHEMATICS 



EXERCISE 25 

1. Construct a triangle each of whose sides is 8". 

2. The sides of a triangle are 3", 4", 5", respectively. 
Construct the triangle. 

3. The sides of a triangle are 13', 15', 16', respectively. 
Draw this triangle to a scale, using i" = 1' — 0". 

4. Draw three altitudes of the triangle constructed in 
example 3. 

5. Construct a, triangle whose sides are 15', 16', 28'. Draw 
the three altitudes. 

6. Measure the three altitudes of the triangle constructed 
in example 1. How do they compare? Is this comparison 
the same in altitudes of other triangles of this exercise ? 

7. In the figure ABCD, in A ABD 
and CDB, x = y and o = z. Compare 
the triangles. Give reasons for your 
conclusion. 

8. In the figure ABCD, in triangles 
ABO and CDA, x is equal to y, o is 
equal to z, Z D is equal to Z B. Com- 
pare the triangles. Give reason for 
your conclusion. 






9. In triangles ABD and CDB, 



Zx = Z z, and Z y = Z o. 
triangles. Why ? 

10. If in triangle ABC, b = a, and CD is 
so drawn that AD = DB, compare triangle 
ADC with triangle CDB. 

11. On a six-inch base construct a triangle 
whose remaining sides are each 14". What do A 
you know about this triangle ? 



Compare the 




LINES, ANGLES, TRIANGLES 85 

12. The sum of two numbers is 24. One number is four 
more than the other. What are the numbers ? (See example 
10, exercise 8.) 

13. The difference of two numbers is 8, and their product 
is 160 more than the square of the less number. Find the 
numbers. 

14. The sides of a right triangle (§ 84) are as 4 to 3* The 
square of the hypotenuse is equal to the sum of the squares of 
the other two sides. The hypotenuse is 25. Find the sides 
and the area. 

15. In a lever AM, P= power; F = fulcrum, on which the 
lever rests ; W= weight. Note that if the 

lever is pressed downward at P, W tends ffi ^ 

to rise. It has been found that : power M2 L 0^ 6-0" A 
times the distance, from where the power is 
applied, to the fulcrum is equal to the iveigJit times the distance 
from the weight to the fulcrum, or 

P. AF=W-MF. 

AF is called the power or force arm, MF, the iceight arm. 

If the power arm is 6' — 0", the weight arm 2 r — 0", and 
the weight 200 pounds, what power is necessary to lift the 
weight ? 

Note. In these problems, the weight of the lever is neglected. 

16. With a 4" x 4", 14 feet long, used as a lever, a 100- 
pound boy on the end of the power arm 
is just able to lift a weight W, when 

~^Zq"— the fulcrum is 2' — 0" from the weight. 

What weight does he lift ? 

* In such statements fractions are avoided by letting the required values be 
represented by a multiple of the unknown number. For example : Let 3s be 
one side, and 4s the other. 




86 



MATHEMATICS 





17. A crowbar is 5' — 0" long. The 
fulcrum is 3" from one end. A force of 
20 pounds at the other end of the bar is 
required to pull a railroad spike from a tie. 
With what force does the spike hold ? 

18. With the crowbar and fulcrum as 
in example 17, two boys weighing 100 
lb. and 115 lb., respectively, were able ^ 
to move a freight car standing on a sid- 
ing. What force was necessary to start 
the car? 

19. The length and breadth of a rectangular tank are as 4 to 
3. The water in the tank is frozen to a depth equal to one 
fourth the width of the tank. The area of the bottom of the 
tank is 48 square feet. Find the weight of the ice. (Ice 
weighs 57J pounds per cubic foot.) Which is heavier, water 
or ice ? Why is this necessary ? See exercise 22, example 3. 

20. Five lines are drawn from a point forming 
_ angles a, 6, c, c?, e. The sum of a and b is 90° ; d is 
/ e equal to 12° more than a; c is equal to &; e is equal 

to twice d. Find the angles. (See § 83, d.) Is the drawing 
correct ? 

21. Divide 48 into three parts, such that the first part shall 
be twice the second, and the third 8 more than the first. 

22. A rectangle and a square have the same altitude. The 
base of the rectangle is 8' more than the base of the square. 
The areas differ by 64 square feet. Find the dimensions of 
each. 

89. An acute angle is an angle that is less than a right 
angle. Ex. Angle a. 

An obtuse angle is an angle that is greater than a 
right angle and less than two right angles. Ex. 
Angle b. 





LINES, ANGLES, TRIANGLES 87 

90- When two straight lines intersect, the opposite angles 
formed are called vertical angles. 

Thus, a and c are vertical angles, also b and d. 

91. An angle is measured by comparing it with 
another angle considered as the unit of measure. 
The most general unit of measure is -^ of a right 
angle, and is called a degree. 

The degree is divided into 60 equal parts called minutes, 
and the minute into 60 equal parts called seconds. 

The abbreviations for degrees, minutes, seconds, are °, f , ", 
respectively. 

92. The right angle (§ 80) is measured by one fourth of the 
circumference of a circle, or 90°. 

^ 93. If the sum of two angles is a right 

angle, each is a complement of the other, 
and the angles are said to be complementary. 
Thus, if ABC is 90°, a is the complement 
"A °^ ^? their sum being 90°. 




94. If the sum of two angles is 180° or two right angles, 
each is the supplement of the other, and the angles are said to 
be supplementary. 

Thus, if the sum of angles c and d is 180°, c 
d/ y£ and d are supplementary and each is the sup- 

plement of the other. 

95. It follows from § 94, that if two supplementary angles, 
as c and d in the figure, are adjacent (§ 72), their exterior sides 
lie in the same straight line. 

Then c and d are supplementary adjacent angles. 

EXERCISE 26 

1. Find the complement of 38° ; the supplement of 38°. 

2. Are 36° 30' and 53° 30' complementary ? Why ? 



88 MATHEMATICS 

3. Are 130° 19" and 49° 41' supplementary ? Why ? 

4. Find the complement of each of these angles : 24° ; 36° ; 
72°; 60°; 30°; 45°; 30° 15'; 24° 15' 20"; 47° 10.5' ; b°. 

5. Find the supplement of each of these angles : 22° ; 96° ; 
120°; 150°; 24° 8'; 27° 19' 36"; 45° 4.4'; 90°; b°. 

6. Find the complement of the supplement of : 120° ; 130° ; 
135° ; c°. Does this last result give a simple formula ? 

7. Find the supplement of the complement of: 3°; 24°; 
28° 5' 30"; 45°; 90°; c°. Does this last result afford a means 
of simplifying other parts of this example ? Illustrate. 

8. Angle at A = 36°, Z£ = 74°. Find the supplement of 
their sum. 

9. Angle at A = 29°, Z B = 31°. Find the complement of 
their sum. 

10. Angle at A = 75°, Z B = 15°. Find the complement of 
their difference. 

Theorem V 

96. Any side of a triangle is greater than the difference of the 
other two sides. 

Draw any triangle ABC, the sides opposite A, B, C, being 
a, b, c, respectively. 

We have 

Given a any side of A ABC, and c> b. 
To prove a> c — b. 

Proof. 1. a + b>c. 

A straight line is the shortest distance between two points. 
(§ 75, Ax. 10). 

2. a >c — b. (Transposing &, § 49.) 

3. Hence, any side, a, is greater than the difference of the 
other two sides. 



LINES, ANGLES, TRIANGLES 89 

Theorem VI 

97. The sum of two sides of a triangle is greater than the sum 
of two lines drawn from any point within the triangle to the 
extremities of the third side of the triangle. 

Draw* any triangle ABC. 

From any point P, within the triangle draw PB and PC. 

Call side AB, c; AC, 6; PB, e ; PC, d. 

We now have : 

Given A ABC, with lines d and e drawn from any point P 
within the triangle. 

To prove c + b > d + e. 

Proof. 1. Produce BP to E.-f 

Call PE, o; AE, x; EC,y. 

2. c -+- x > e + o. 

A straight line is the shortest distance between two points 
(§ 75, Ax. 10). 

3. o + y > d (§ 75, Ax. 10). 

4. Add 2 and 3, 

c + x + o-\-y>d + o + e. 

5. Subtract o from each side of this inequality, 

' c + x + y>d + e (§46). 

6. But x + y = b (§ 23, Ax. 6). 

7. Whence, substituting for x + y its equal 6, 

c + & > d + e. 

* This description of the drawing is not a part of the demonstration. Your 
demonstration depends solely on the three paragraphs: Given, To prove, 
Proof. Be sure you know what you have given you to work with and what 
you wish to prove. 

t Remember that any additional lines you draw in a figure must be dotted 
lines. 




90 MATHEMATICS 

Theorem VII 

98. If two straight lines intersect) the vertical angles are equal 

Draw two intersecting Y 

lines, MN and X Y, form- 
ing the angles a } b, c, d. 
We then have 

Given two intersecting 
lines, MN and XY, and 
vertical angles a, c, and b, d. 

To prove a = c and b = d. 

Proof. 1. a + b = 2 rt. A (§ 95). 

2. b + c = 2 rt. A (§ 95). 

3. Subtract 2 from 1 (§ 23, Ax. 2), 

a — c = 0, 
or a = c. 

4. The pupil may prove b = d. 

What is the hypothesis in the above theorem ? What is 
the conclusion ? 

Problem 

99. To draw a perpendicular bisector of a straight line. 

Draw line AB. 

To draw a perpendicular bisector of AB, 

1. With A as center and any radius greater than one half 
AB, describe arcs on each side of line AB. 

2. With B as center and the same radius, describe arcs in- 
tersecting the arcs already drawn at and at D. 

3. Draw CD. 

4. Then CD is the perpendicular bisector of AB. 

The reason for this construction will be given in § 102. 
F, the point of intersection of CD and AB, is called the foot of 
the perpendicular. 

Note that C and D are each equally distant from A and B. 



LINES, ANGLES, TRIANGLES 91 

Problem 

100. To draiv a perpendicular to a line at any point in 
the line. 

1. Draw a line AB. 

2. Take any point P in the line AB. 

3. To erect a perpendicular to line AB at P. 

4. With P as center and any radius, describe arcs inter- 
secting AB, or AB produced, at C and E. 

5. With and E as centers and radius greater than one 
half CE, describe arcs intersecting on one side of AB. 

6. The pupil may show that the perpendicular may now 
be drawn and that this problem is an application of § 99. 

Why must the radius be greater than one half CE ? 

EXERCISE 27 

1. How many points determine a line? What do you mean 
by determine as used in this sense ? 

2. How many conditions can be imposed on a line ? What 
properties besides length has a straight line ? 

Theorem VIII 

101. If a perpendicular is erected at the middle point of a line, 

I, Any point in the perpendicular is equidistant from the ex- 
tremities of the line. 

II. Any point not in the perpendicular is unequally distant 
from the extremities of the line. 

Draw line KF A. line AB at its middle point F. 
From P, any point in KF, draw lines PA and PB. 
We then have, 

I. Given two lines PA (x) and PB (y) drawn from any 
point P in the perpendicular to AB at its middle point. 

To prove ® = y> 



92 



MATHEMATICS 



Proof. 1. In triangles AFP and PFB, call AF = o, FB = b. 

o = b (Constr.) 
h = h (Identical) 
ZAFP=ZBFP. 

(All rt. A are equal.) 

2. Then, AAFP=APFB. 

(Two triangles are equal when two sides, etc., § 77.) 

3. Then, x = y. 

(In equal figures corresponding parts are equal.) 

Does this remind you of Theorem III? 

II. Given P ' any point not in KF. 

To prove PA =£ PB* 

or x-\- c^=m. 

Proof. 1. If P is not in KF, P'A or 
P'B must intersect KF. 

Suppose PA intersects KF at P. 

Draw PB (y). 

2. Then, y + c^m (§ 75, Ax. 10). 

3. But ' y = a (§101,1). 

4. Then, substituting x for its equal y in 2, 

# + c =£ m. 

102. From the result of § 101, we may conclude that : 

I. A point equally distant from the extremities of a line lies 
in the perpendicular at the middle point of the line. 

II. Two points each equally distant from the extremities of a 
line determine the perpendicular at the middle point of the line. 

III. Two lines drawn from a point in the perpendicular to 
a line and cutting off equal distances from the foot of the per- 
pendicular make equal angles with the perpendicular, and are 
equal. (The proof of III is left to the pupil.) 

* ^ is read "does not equal. 11 




LINES, ANGLES, TRIANGLES 93 

Problem 

103. To draw a perpendicular to a line from a given point 
ivithout the line. 

Given line AB and point P without AB. 
To draw PF± AB. Construction : 

1. With. P as center and a radius greater than the distance 
from P to line AB, describe an arc intersecting AB at E and D. 

2. With E and D as centers and the same radius greater 
than one half ED, describe arcs intersecting at K. 

3. Draw PA". 

4. PK is the required perpendicular. 

Note that P and K are each equally distant from E and D. 
Compare EF and ED (§ 101). 

Theorem IX 

104. From a point without a line but one perpendicular can be 
drawn to the line. 

Draw line AB and PF1.AB, also PH any other line meet- 
ing AB at H. We have 

Given PF± AB, and PH any other line from P to AB. 

To prove PH not perpendicular to AB. 

Proof. 1. Produce PF to P' making FP = PF. 

2. Draw HP'. 

3. Eepresent A PHF and PHF by x and y, respectively. 

4. Then, Ax = Ay{% 102, III). 

5. PFP' is a straight line (by construction). 

6. Then, PHP' is not a straight line. 

(But one straight line can be drawn between two points, Ax. 9.) 

7. Then, x + y^lSO. (§95) 

8. Whence, 2 x =£ 180, and a #= 90. 

9. Hence, Pi?" is not perpendicular to AB. And since PH 
is any line except PF, PF is the only perpendicular that can 
be drawn from P to line AB. 



94 



MATHEMATICS 



EXERCISE 28 

1. The base of an isosceles triangle is 8", the altitude &'. 
Construct the triangle. 

2. The altitude of an isosceles triangle is 8", one leg is 10". 
Construct the triangle. 

3. The sides of a triangle are 5", 12", 13", respectively. 
Construct the triangle. 

4. The base of a triangle is 13", the base angles each 45°. 
Construct the triangle. Measure the angle at the vertex.* 

5. The sides of a triangle are 4", 10", 6", respectively. 
Construct the triangle. Explain your result. 

6. Two sides and the included angle of a triangle are 8", 
12", 45°, respectively. Construct the triangle. 

7. Draw two acute angles, a and b. At a given point on a 
line AB, construct an angle equal to the sum of these two 
angles. 

8. Draw a triangle ABC, and at a given point P, on a line 
DE, construct an angle equal to the sum of the angles at A, 
B, and C. 

Theorem X 

105. Two right triangles are equal when the hypotenuse and an 
acute angle of the one are equal, respectively, to the hypotenuse 
and acute angle of the other. 



XC 





Draw right A ABC, and side A'B' (c') = side AB (c). 
* The vertex opposite the base is the vertex of the triangle. 



LINES, ANGLES, TRIANGLES 



95 



At B' construct Z B' = Z B, and draw A'C _L B'E. Repre- 
sent BC, B'C by a and a', respectively, also AC, A'C 1 by b 
and 6', respectively. 

We then have 

Given rt. A ABC and A' B'C with c = c', and Z .B = Z £'. 
To prove A ABC = A J/.B'<7 r . 

Proof. 1. Place A JJ3(7 on A A' B'C in such a manner that 
c will coincide with its equal c'. 

2. Since ZB = ZB', a will fall on a'. 

3. A falls on A', then side & will coincide with side V. 
(But one _L can be drawn from a point to a line from a 

point without the line, § 104.) 

4. Hence, the triangles are equal, since they coincide 
throughout. 



\\ 



Theorem XI 

106. Two right triangles are equal when the hypotenuse and a 
leg of the one are equal, re- 
spectively, to the hypotenuse *> 
and a leg of the other. 

Draw rt. A ABC On 
line DH, at G, erect EG _L 
DH and equal to b. With 
E as center and c as radius, 
describe an arc intersect- 
ing DH at F. We then D 
have 

Given two right trian- 
gles, ABC and EFG, with 
c—g, and b =f E 

To prove A ABC = A £^#. 

Proof. 1. Apply A £^(2 to A ABC in such a manner that 
/will coincide with its equal b, vertex F falling at FJ 




96 



MATHEMATICS 



2. Since A ACB and ACF' are right angles, BCF' is a 
straight line (§ 83). 




3. Since c = g A BAF 1 is isosceles, and Z B = Z J" (§ 85). 

4. Then, A ABC = A ACF ! (§ 105). 

5. Then, A ABC = A EFG. Why? 

Theorem XII 

107. If two unequal oblique lines drawn from a point in a 
perpendicular to a line, cut off unequal distances from the foot 
of the perpendicular, the more remote is the greater. 




Draw line AB, and PF± AB at F, also lines PD and PC, 
DF being greater than CF. 

To prove PD > PC. 

Extend PF to P\ making FP' = PF. 
Draw DP and CP. 

The proof is left to the student (§ 97). 









LINES, ANGLES, TRIANGLES 97 

Theorem XIII 

108. If oblique lines are drawn from a point to a straight line 
and a perpendicular is drawn from the point to the line, 

I. Two equal oblique lines cut off equal distances from the 
foot of the perpendicular. 

II. TJie greater of two unequal oblique lines cuts off the greater 
distance from the foot of the perpendicular. 

1. Call AB the given line, P the given point, PF the per- 
pendicular to AB, PE and PC the equal oblique lines. 

The proof is left to the student. 

(Prove A PFE = A PFC.) 

II. Draw PF± AB, also lines PC (x) and PD (y), with x > y. 

We then have 

Given PF± AB, and line x > line y. 

To prove CF > DF. 

Proof. 1. If CF=DF, x = y (§102, III). But this is 
contrary to hypothesis. 

2. If CF<DF, x<y (§107). This is also contrary to 
hypothesis. 

3. Hence, if CF is not equal to or less than DF, it must be 
greater than DF. 

This manner of proving is called "The Indirect Method." 
What is the hypothesis ? What the conclusion ? Compare 
§ 102, III. What is the relation between the hypothesis of 
§ 102, III, and the conclusion of § 108, I ? When the conclu- 
sion of one theorem is the hypothesis of the other, and the 
hypothesis of the one is the conclusion of the other, each 
theorem is the converse of the other 

EXERCISE 29 

1. The line joining the vertex of an isosceles triangle to the 
middle point of the base bisects the vertical angle. 



98 MATHEMATICS 

2. Two isosceles triangles have a common base but unequal 
altitudes. Show that the line connecting their vertices is per- 
pendicular to the base and bisects the base. 

3. If the altitude of a triangle bisects the base, the triangle 
is isosceles. 

4. Two isosceles triangles are equal if a leg and the vertical 
angle of one are equal, respectively, to a leg and the vertical 
angle of the other. 

5. The hypotenuse of each of two right triangles is 81' — 6". 
Each has an angle of 31° 41'. Compare the triangles. 

6. The hypotenuse and one leg of a triangle are 3 yards, 
1 foot, and 1 yard, 2 feet, and 6 inches, respectively. The 
hypotenuse and one leg of another triangle are 10' — 0" and 
5' —6", respectively. Compare the triangles. 

ORAL REVIEW 
Give the results of the following : — 

1. 19-17. 8. 6.11-7.11. 

2. 16-15. 9. 9-15 + 6.15. 

3. 18 • 16. 10. 8 • 17 + 7 • 17. 

4. 18.5-6-17. 11. 8-16 + 8-16. 

5. 24.6 + 3-17. 12. 27-3-4-28. 

6. 4-18-3-16. 13. 10-34-5.17. 

7. 24 . 5 + 3 • 17. 14. 12 • 8 + 12 . 8. 

WRITTEN REVIEW* 

1. Two lines AB and CD intersect in such a way that they 
bisect each other. If the extremities of these lines are joined 
in order, prove that the opposite sides of the quadrilateral thus 
formed are equal. 

* The teacher will find it advantageous to use some of these before all 
thirteen theorems are proved. 



LINES, ANGLES, TRIANGLES 99 

2. The bisectors of the three angles of an equilateral tri- 
angle meet in a point, thus forming three small A. Prove 
'these A equal. 

3. In an isosceles triangle a perpendicular is dropped from 
the vertex angle to the base. Prove that the perpendicular 
bisects the base. 

4. The opposite sides of a quadrilateral are equal. Prove 
the diagonal divides quadrilateral into two equal A. Also 
prove opposite A of quadrilateral are equal. 

5. Two straight lines intersect in such a way that the dif- 
ference between two adjacent angles is 42°. Find number of 
degrees in each of the four angles formed. 

6. Two lines, a and b, intersect at D forming two acute A 
x and y, and two obtuse A z and iv. At D, a _L is erected to 6, 
dividing Z z into two parts v and t, so that Z t is 24° 30'. Find 
number of degrees in A x, y, z, w. 

7. The middle point of the base of an isosceles A is joined, 
by lines, to the middle points of the legs. Prove the A formed 
are equal. 

8. The vertex angle of an isosceles triangle is bisected. 
Prove the bisector of the angle bisects the base at right angles. 

9. Prom the middle point of the base of an isosceles tri- 
angle perpendiculars are drawn to the legs. Prove the per- 
pendiculars are equal. 

10. Two lines, AB and CD intersect in such a way that AB 
is bisected. Prom extremities A and B, perpendiculars are 
drawn to line CD. Prove these perpendiculars are equal. 

11. Prom the vertex of an isosceles triangle a perpendicular 
is drawn to the base. Prove triangles formed are equal. 

Supplemental Applied Mathematics 

1. A grown person needs 3000 cu. ft. of air per hour that 
the functions of the body may be active. If a room 20 ft. by 



100 MATHEMATICS 

15 ft. by 10 ft. were occupied by one person, how often would 
the air have to be completely changed to obtain pure air ? 

2. How often would the air in a room 12 ft. by 11 ft. by Si- 
ft, have to be completely changed to obtain pure air for two 
persons ? 

3. If there were 500 people in a lecture hall, how often 
would the air have to be completely changed to insure good 
ventilation ? Nine square feet of floor space is allowed to each 
person, and the hall is 11 feet high. 

4. For good ventilation the air in a room containing one 
person needs to be changed every 45 minutes. What is the 
approximate size of the room ? 

5. It has been found by experiment that the air in a room 
cannot be changed more than three times per hour without 
danger of drafts. What are the dimensions of a room that is 
just large enough to meet this ventilating requirement, when 
one person occupies the room ? 

6. It has been found by experiment that one gas jet, when 
burning, uses as much air as two persons. By changing the air 
in a room occupied by one person once an hour during the 
daytime, good ventilation is secured. How many times per 
hour does it need to be changed in the evening when one gas 
jet is burning ? 

7. When two persons occupy a room, good ventilation is 
secured in daytime by changing the air once each half hour. 
How often should the air be changed when 3 gas jets are 
lighted ? 

8. The air in a room occupied by one person needs to be 
changed once every hour in the daytime and three times every 
hour during the evening. How many gas jets are burning ? 

9. A kerosene lamp requires as much air as 4 persons. 
When 2 persons occupy a room, the air needs to be changed 
once every 2 hours. When a lamp burns in the room, how 
often does the air need to be changed ? 



LINES, ANGLES, TRIANGLES 101 

10. In the daytime the air of a room containing 2 persons 
needs to be changed once every 45 minutes, and in the evening 
once every 15 minutes. The room is lighted by kerosene lamps. 
How many kerosene lamps are burning in the room ? 

11. The quantity of carbon dioxide given off by candles is 
about twice as much as that given off by gas. If the air in a 
room needs to be changed once every 1± hours when illuminated 
by gas, how often will it need to be changed when illuminated 
by candles ? 

12. For practical purposes architects figure 30 cubic feet of 
air per minute for each person. A classroom has a ventilating 
system. The room is 28' X 23' X 15' and contains 30 pupils. 
To insure good ventilation, how much air must be driven into 
the room and how many times per hour must the air be 
changed ? 

13. In hospitals it is customary to allow 50 cubic feet of air 
per minute per person. In a hospital ward 56' X 9' X 12' are 8 
patients, a nurse, and 2 gas jets. How much air must be sup- 
plied per hour ? How many times must it be changed per 
hour ? 

14. A train running 40 miles an hour strikes two torpedoes 
400 feet apart. Sound travels 1090 feet per second. What 
time elapses between their reports at a station which the train 
is nearing ? 

15. According to some engineers, the sectional area of the 
cold-air box of a furnace should be equal to the combined areas 
of all the registers. There are 6 registers in a house, each 8 in. 
by 10 in. How large should the cold-air box be ? 

16. The cross section of a cold-air box is 2 ft. 3 in. by 1 ft. 
8 in. There are 6 equal registers in the house. What is the 
area of each register ? 

17. The sectional area of a cold-air box is 495 sq. in. Each 
register measures 9 in. by 11 in. How many registers are 
there in the house ? 



102 MATHEMATICS 

18. According to other engineers, the sectional area of a cold- 
air box should be equal to the combined areas of all the registers 
minus one sixth. There are 8 registers in a house, each 8 in. 
by 10 in. How large should the cold-air box be ? 

19. The sectional area of a cold-air box is 600 sq. in. ; each 
register measures 8 in. by 10 in. How many registers are there 
in the house ? 

20. The cross section of a cold-air box measures 20 in. by 20 
in. There are 6 equal registers in the house. What is the area 
of each ? 

21. It is said that the minimum dimensions of an ideal din- 
ing room are 11 ft. by 13^ ft. How many square feet are con- 
tained in such a room ? 

22. An ideal dining room of maximum size measures 17 ft. 
by 22 ft. How many square feet are contained in such a room ? 

23. A kitchen, according to one authority, should measure 
10 ft. by 12 ft. If the range measures 2 ft. by 3 ft. 14 in., the 
sink 1 ft. 6 in. by 3 ft. 15 in., two cupboards each 1 ft. 8 in. by 
6 ft. 4 in., and the work table 4 ft. by 5 ft. 7 in., how many 
square feet are allowed for " walking " space ? 

24. It is said that stairs are well proportioned when 2 times 
the height of the riser, added to the tread, equals 24 in. The 
riser of such stairs is 7 in. What is the tread ? 

25. Measure the stairs in your home. How near do they 
come to the ideal measurements ? 

26. How large a piece of material must I have to make a bag 
10"xl4" when finished, if I allow 2" for a heading and V' on 
three sides for hems ? Use the width of the material for the 
depth of the bag. 



CHAPTER VII 

Graphs, the Algebra of Lines. Parallels and their Uses 

109. In §§ 50-56 we studied and solved simultaneous equa- 
tions. We will now study these equations from a geometric 
standpoint. 

In § 25 we determined that if toward the right were posi- 
tive, toward the left should be negative. Suppose we 

measure our positive and nega- 
tive values from two lines inter- 
secting at right angles. Suppose 
further, that horizontal measure- 
ments shall be ^-measurements 
-X and vertical measurements be 
^/-measurements, upward being 
positive and downward being neg- 
ative. 



X 







31 



Y f 



Ex. 1. Eind the point where x = 2, y = 3. 

Measuring OM = 2 and MP = 3, we have the point T satisfying the 
required condition. 



EXERCISE 30 
In the same manner locate the following points : 

1. x = 4, y = 2. 5. x = — 1, y = — 4. 

2. x — 3, y — 5. 6. x= — 1, 2/ = 2. 

3. x = 2, y = — 3. 7. x = 5, y = — 1. 

4. x = — 1, y = 4. 8. x = 0, y = 2. 

103 



104 



MATHEMATICS 



9. x = 2, y = 0. 

11. x = — 2, y = 0. 



10. a; = 7 ?/ = — 3. 



12. Are these points on the same circumference ? 

a? = 3, # = 4 ; x = A, y = 3; x= — 3, ,y = 4 ; 

aj = 4, ?/ = — 3 5 ;r=-4 ? ;?/ = 3; g = 3, y = — 4 ; 
x = - 3, y = - 4 ; aj = - 4, y = - 3. 

(Take your center at 0.) 

110. In the equation x + y = 4t, x and y are so related that 
their sum must always equal 4. x and y then may take any 
values whose sum is 4. To find such pairs of values, solve 
x + y = 4 for either # or y. We will solve for y. 



y 



x. 



Give x a set of values, say 0, 1, 2, 3, etc., and find the corresponding 
values ofy. If x = 0, y = 4 ; if x = 1, y = 3, etc. 

It is convenient to write these values in two columns headed x 
and ?/. See below. 






4(^) 


1 


3 (2?) 


2 


2 (C) 


3 


1 (D) 


4 


(#) 


5 


-1 (F) 


-1 


5 etc. 




Now plot these values on the same pair of axes, just as we 
did in exercise 16. 

In this way we obtain the points A, J5, (7, Z>, etc. 

If these points are connected by a smooth curve, any point on the 
curve will correspond to a point satisfying- an x and a y of x + y = 4. 



GRAPHS, THE ALGEBRA OF LINES 105 

The values of x and y which locate the position of a point 
are called coordinates of a point. The ^-measurement, e.g., OM, 
is called the abscissa, the ^/-measurement, e.g., MB, is the ordi- 
nate. The point is called the origin. 

If each term of the equation is of the first degree, the 
curve ABCDEF is a straight line and the equation is called 
linear. 

This curve is called the graph or geometric picture of the 
equation. 

111. The degree of a term is determined by the sum of the 

exponents of the letters in it. In an equation the term of 
highest degree determines the degree of the equation. 

x 3 y + xy 2 — x 3 is an expression of the fourth degree in xy and 
of the third degree in x. 

x 3 — 3 x 2 = 8 is an equation of the third degree. 

2 x + a 2 = b 3 — c is an equation of the first degree in x. 

EXERCISE 31 
Find the graphs of the following equations : 

1. x +y = 6. 5. 2x — y — 5. 

2. 2x+y = 8. 6. 3x-2y = 5. 

3. x — y=&. 7. x + 4?/ = —3. 

4. 4 a — 3y = 12. 8. x+2y = — 4. 

112. If a pair of simultaneous equations (§ 50) are plotted 
on the same axes, their graphs will usually intersect. In this 
case the coordinates of the point of intersection are the same 
as the values that are found for x and y when the equations 
are solved. 

This explains in a geometric way the name simultaneous 
equations. The x and y must at the same time have values 
which satisfy both equations. 



106 



MATHEMATICS 



Ex. Solve by graphs : 
x — 2y = — 5, 
x + y=l. 



(1) 
(2) 



From (1) 
x 



1 

2 
-1 



y 
+* 

+ 3 

l 

2 



From (2) 
x y 





1 

2 

-1 




In the above problem the lines 
(1) and (2) intersect at % — — 1, ?/=2. If the equations are solved 
by the principles of §§ 53-55, the same results are obtained. 

113. Note that in these equations x and y vary. Any 
change in one of these numbers causes a change in the other. 
For this reason x and y are the variables in the equation. (Com- 
pare § 51.) Letters and numbers whose values do not change 
in an equation are called constants. The number which is inde- 
pendent of the variables is called the absolute term. 

Sometimes the graphs of equations will not intersect. The 
equations are then said to be inconsistent. 

Ex. Plot the equations x + y = 4, 

a? + y = 2 . 

The graphs of these equations do not intersect. This is be- 
cause the lines are parallel. 

114. Parallel Lines. Two straight lines are said to be parallel 
when they have the same direction. It is evident that if they 
are drawn through two different points they are everywhere 
equally distant and can never meet. 

115. Some of the graphs of equations do not intersect because 
they coincide. 



GRAPHS, THE ALGEBRA OP LINES 107 

Ex. Plot 2x + 3y = 3, 
4zX + 6y = 6. 

These graphs coincide. Equations of this kind are said to be 
equivalent. See example 2, exercise 22. 

Equivalent equations and inconsistent equations have no 
solution. 

EXERCISE 32 

Solve by means of graphs. In each case solve algebraically, 
also compare the result with the graphical solution. Note that 
if the roots are fractional, the graphical solution is only ap- 
proximately correct. 

1. x + y = 7, 5 x_y ==2 7. 4a>-7# = 18, 
2x-3y=-6. "2 4 6^ + 5y=-4. 

2. Sx + 42/ = 11, 3#_2/ = !5 8. 9a?-4y = l, 
4 s-3,7=-2. 4 3 6' 6a? + 2y = 3. 

3. ^ + 2?/= 13, 6. 5x + 8?/ = 13, 9. 5x -4y = 2, 

2 a?- y = l. ^ 6a?- 5y = l. 4a; + 5y=-2. 

4. a + 2y = 5, 

2x -h y = 4. 

What angle is formed by the lines in examples 2, 3, 9 ? 

10. 2 a — 5 ?/ = 3, 12. 2 a + y = 4, 14. 4 tc — 7 v = — 3, 

3x — l\y=4c\. 2x + y = 6. w + 3 v = 4. 

11. 8x + 2y = l, 13.5x-3y = 4: 9 

Ax + y — \. 5 x — 3 y = 5. 

What relations exist between the three lines : 

15. 4x + 3y = 2, 

3# — 4^ = 5, 
3x-4,y = 8. 



108 MATHEMATICS 

16. 5 t + u = 11, 17. 2 m + 3 fc = 19, 

3t-2u = 4. 3 m - 4 A; = 3. 

18. 6 v — 8 w = 5, 
4 y + 5 w = 7. 



Parallels and their Uses 

116. Parallel lines (§ 114) are of great importance in 
geometry. 

117. We assume that but one line can be drawn through a 
given point parallel to a given line. 

118. Prove that in the same plane two perpendiculars to the 
same line are parallel. 

Hint. Draw the line x -f y = — 4. At points x = — 4, y = 0, and 
x = 0, y = — 4, erect perpendiculars to this line. To prove that these 
perpendiculars are parallel, suppose that they meet if produced, then 
read Theorem IX. The method of proof you use here is called Indirect. 

(§ 108.) 

Compare your given equation, x — y = — 4, and x — y = 4, 
with the three lines of your figure. Are similar relations 
found in exercise 32 ? 

Problem 

119. To draw a straight line parallel to a given straight line. 

1. Draw an indefinite line AB. 

2. Choose a point P without line AB through which to 
draw the parallel. 

3. Draw AP and extend it to some point K. 

4. At P construct an angle equal to /LA. 

5. PZ, one side of Z. ZPK is the required parallel. 



PARALLELS AND THEIR USES 109 

Theorem XIV 

120. Two lines parallel to the same line are parallel to each 
other. 

Draw line a. Draw lines b and c parallel to a. We then have 

Giver two lines b and c parallel to line a. 

To prove line b || line c. 

Proof. 1. If b and c are not parallel they will, if produced, 
meet at some point P. 

2. The rest of the proof is left to the pupil. (See § 117.) 

Theorem XV 

121. A line perpendicular to one of tivo parallels is perpendicu- 
lar to the other. 

Draw line a || line b. Draw line c _L line b, intersecting line 
a at K. We now have 

Given lines a and b parallel, and line c _L line b. 

To prove line c _L line a. 

Proof. 1. Draw line m through K perpendicular to c. 

2. Then, lines b and m are ± to c, and b and m are ||. (§ 118.) 

3. Then, m and a coincide. (§ 117.) 

4. Hence, c _L a. 

122. The angle opposite the base of a triangle is the vertical 
angle. (Any side may be the base.) 

The vertex of the vertical angle is the vertex of the triangle. 

The altitude of a triangle is the perpendicular from the vertex 
to the base. 

An exterior angle of a triangle is the angle formed by any 
side of a triangle and the adjacent side produced. 

EXERCISE 33 

1. In an isosceles triangle draw the exterior angle at the 
base : at the vertex. 

2. Draw three altitudes of an isosceles triangle. 



110 



MATHEMATICS 



3. Draw three altitudes of an equilateral triangle. 

4. Draw three altitudes of an obtuse angled triangle. 

5. Draw three altitudes of a right triangle. 

6. Draw the three bisectors of the angles of the triangles in 
examples 2 to 5. 

7. Do the three altitudes of these triangles ever meet in a 
point ? Do they always meet in a point ? 

8. Do the altitudes and the bisectors ever coincide ? If so, 
when? 

9. Do the three bisectors ever meet in a point ? Do they 
always meet in a point ? Later in the course you will be 
called upon to prove your above conclusion. 

123. If two lines AB and XFare cut by a third line MN, 
MN is said to be a transversal. 

This transversal makes with the other two lines eight angles 
which have special names, 
names which refer to the 
position of the angles 
with respect to the lines. 
For example, a, b, o, s, 
are between lines AB 
and XY, and are called 
interior angles. The re- 
maining four are exterior 
angles. 

a and s being on oppo- 
site sides of the trans- 
versal are alternate-interior angles, 
alternate-interior angles. 

c and s, on the same side of the transversal are exterior- 
interior angles. Locate the other exterior-interior angles. 

d and g are alternate-exterior angles. Find the other alter- 
nate-exterior angles. 




Likewise o and b are 



PARALLELS AND THEIR USES 111 

Theorem XVI 

124. If two parallel lines are cut by a transversal, the alternate- 
interior angles are equal. 

Draw AB II XY, and 30T intersecting AB and XFat and 
K, respectively. Let Z KOA = c, and Z YKO = d. We 
then have 

Given lis AB and XY cnt by MN forming alternate-interior 
angles c and d. 

To prove Z c = Z d. 

Proof. 1. Through Z, the middle point of OK, draw a _L to 
^J3, meeting ^45 at F, and XF at E. Let Z FZO = g and 

z ^zs = *. 

2. EF±XY. (§ 121.) 

3. In rt. A OFZ and Ji^Z, 

g = i- (§98.) 

JTZ = ZO. (Constr.) 

4. Hence, A O.FZ= A KEZ. (?) 

5. Then, Z c = Z d. (§ 86.) 

Tliis is the fundamental proposition in parallel lines. 

EXERCISE 34 

1. If two parallels are cut by a transversal, the exterior- 
interior angles are equal. (Use the equations derived from the 
Theorems XVI and VII.) Note : Remember in proving any 
geometric statement you must give authority (the why) for 
each step you take. 

2. If two parallels are cut by a transversal, the sum of the 
interior angles on the same side of the transversal is equal to 
two right angles. 

3. If two parallels are cut by a transversal, the alternate- 
exterior angles are equal. 



112 MATHEMATICS 

4. If two parallels are cut by a transversal, the sum of the 
exterior angles on the same side of the transversal is equal to 
two right angles. 

Theorem XVII 

125. If two lines are cut by a transversal, and the alternate- 
interior angles are equal, the lines are parallel. 

Draw a line AB. Draw a line MN intersecting AB at 0. 
Through any point K on MN, draw line XY, making an angle 
YKO equal to KOA. Use same notation for angles as that 
in Theorem XVI. We have 

Given two lines AB and XFcut by MN making Z c = Z d. 

To prove AB II XY. 

Proof. 1. Through K draw GH II AB. 

2. Then, Z 0KH=Zc. (Theorem XVI.) 

3. But Zd=Zc. (By Hyp.) 

4. Then, Z 0KH=Zcl. (Ax. 8.) 

5. Hence, lines Oil and XY coincide. 

6. Therefore, XYWAB. 

This theorem is the converse of Theorem XVI. That is, the 
hypothesis and conclusion of the two theorems are inter- 
changed. Theorems XVI and XVII are very important. 

EXERCISE 35 

1. If two lines are cut by a transversal and the exterior- 
interior angles are equal, the lines are parallel. 

2. If two lines are cut by a transversal, and the sum of the 
interior angles on the same side of the transversal is equal to 
two right angles, the lines are parallel. 

3. If two lines are cut by a transversal, and the alternate- 
exterior angles are equal, the lines are parallel. 

4. If two lines are cut by a transversal, and the sum of the 
exterior angles on the same side of the transversal is 180°, the 
lines are parallel. 



PARALLELS AND THEIR USES 113 

5. If two angles have their sides parallel, each to each, they 
are either equal or supplementary. {Hint. Produce one side 
of each angle, if necessary, until the lines intersect.) 

6. If two angles have their sides perpendicular, each to 
each, they are either equal or supplementary. 

Draw A ABC, also A KOM whose sides are perpendicular, 

each to each, to the sides of 
A ABC. We have 

Given A ABC and KOM 
! 7 / with side AB JL KO, and 






D- 



\ 



K 



-} -jy BC±OM. 

To prove AB equal to or 
\r£ supplementary to Z KOM. 

Proof. 1. * Through draw 
DE JL OM, D and E being on opposite sides of ; produce 
KO to G, and draw 0i7 JL GK Call AG0E,d; A HOE, i; 
A M0H,c, A KOM, b; A DOK, a. 

2. Then DE WBG. (?) 

3. And OHWBA. (?) 

4. Therefore Z t = Z JB. (?) 

5. i +c== $o°. (?) 

6. b + c = 90°. (?) 

7. Thent = &. (?) 

8. AndZJS = Z6. (?) 

9. If OKweve drawn in the opposite direction, namely OG, 
A MOG would be the supplement of Z B. 

7. Two triangles have their sides mutually perpendicular. 
Show that they are mutually equiangular. 

8. Two triangles have their sides mutually parallel. Show 
that they are mutually equiangular. Are the triangles equal ? 

* Remember that all additional lines drawn in a figure must be dotted lines. 



114 MATHEMATICS 

Theorem XVIII 

126. The sum of the angles of a triangle is equal to two right 
angles. 

Given A ABC. 

To prove that Z A + Z B + Z ACB = 2 rt, A. 

Proof. 1. Produce side BO to iT. Draw (7iff \\ BA, and on 
same side of BO as J3A 

Let Z.ACB = x, AACM = y, AMCK = z. 

2. x + y + % = 2 rt. A. (§ 83, c.) 

3. AB = z (?) 

4. Z^l = 2/ (?) 

5. a = a (?) 

6. Add equations 3, 4, and 5. 

7. Compare equations 2 and 6. 

8. Hence? 

EXERCISE 36 

1. Prove Theorem XVIII by drawing a line through the 
vertex, parallel to the base. (Do not draw any other lines.) 

2. Prove two right triangles equal if a leg and an acute angle 
of the one are equal, respectively, to the leg and acute angle of 
the other. 

3. Prove that the exterior angle (§ 122) of a triangle is 
equal to the sum of the two opposite interior angles. (You 
will need this theorem very often.) 

4. Prove that the exterior angle of a triangle is greater 
than either of the opposite interior angles. 

5. How many right angles can a triangle have ? 

6. How many obtuse angles can a triangle have? 

7. One angle at the base of an isosceles triangle is 36°. 
Find the other angles. 



PARALLELS AND THEIR USES 115 

8. The vertical angle of an isosceles triangle is 120°. The 
base angles are bisected. Find the angles formed by the 
bisectors. 

9. The angle formed by the bisectors of the base angles of 
an isosceles triangle is 100°. Find the angles of the triangle. 

10. One angle of a right triangle is 45°. Compare the legs 
of the triangle. 

11. (a) Construct an equilateral triangle. 
(b) Construct an angle of 30°. 

12. Construct an angle of 45°. 

13. An exterior angle at the base of an isosceles triangle 
formed by producing the base is 108°. Find the angles of the 
triangle. 

14. Could the exterior angle in example 13 be 89°? 90°? 
91°? Why? 

15. An exterior angle formed at the vertex of an isosceles 
triangle by producing one of the legs is 129°. Find the angles 
of the triangle. 

16. Find the exterior angle at the base of an equilateral 
triangle. ? 

17. Prove 6 of Exercise 35 by producing MO to meet BC, 
and producing BA to meet OK produced. 

127. Develop the following synopsis : 
Two triangles are equal if (a) 

(p) 
oo 

Two right triangles are equal if (a) 

(6) 

00 

id) 

Keep this synopsis always in your mind. 



116 MATHEMATICS 

Theorem XIX 

128. If two angles of a triangle are equal, the triangle is isosceles. 

Draw a line CD. At C and D construct equal angles. Let 
the sides of these angles meet at P. Call PC, d, and PD, c. 
We then have 

Given A CDP with Z G = Z D. 

To prove c = cl. 

Proof. 1. Draw a perpendicular from P to the base. 

2. Prove the A formed are equal. 

Theorem XX 

129. If two sides of a triangle are unequal, the angles opposite 
are unequal, and the greater a)igle lies opposite the greater side. 

Draw A DEF making DE > DF. Call BE, f and DF, e. 
We now have 




M E 



Given A DEF with f>e. 

To prove Z EFD > ZE. 

Proof. 1. On DE take DM = e, and draw JFJf. 

Call Z DTtfi^ a, and Z MFZ), #. 

(Note in this construction that D is the angle not involved 
in the statement, and from D we measure the distance DM.) 

2. x = ?/. (Theorem III.) 

3. ZEFD> y. (Ax. 7.) 

4. Zo/^. (Ex. 36, 4.) 

5. .-. 7/ > Z ^. 

6. ,'.ZEFD>ZE. (?) 



PARALLELS AND THEIR USES 117 

EXERCISE 37 

1. The perpendicular is the shortest line from a point to a 
line. (Draw the perpendicular and any other line from the 
given point to the given line.) 

2. Two isosceles triangles are equal if the base and one 
base angle of one are equal, respectively, to the base and one 
base angle of the other. 

3. The hypotenuse in a triangle is greater than either leg. 

4. Find the sum of the angles of a quadrilateral. (Theorem 
XVIII.) 

5. If the lines are drawn from a point within a triangle to 
the extremities of one side, the angle included by theni is 
greater than the angle included by the other two sides. (Use 
the figure of Theorem VI, and apply Exercise 36, Example 4.) 

Theorem XXI 

130. Any point in the bisector of an angle is equidistant from 
the sides of the angle. 

Draw A DEF. Bisect Z E. 

Erom P, or any point in the bisector, draw PM and PK _L 
ED and EF, respectively, meeting ED at M and EF at K. 
Call PM, d x and PK, d 2 . We now have 

Given ADEF, and PE bisecting DEF, also, d l and d 2 js 
from any point, P, in the bisector, to sides ED and EF, 
respectively. 

To prove d ± — d 2 . 

Proof. Show that A EPK= A EMP. 

131. This bisector is sometimes called the locus of points equi- 
distant from the sides of the angle. A locus may be defined as 
a point or line which fulfills conditions imposed upon it, no 



118 MATHEMATICS 

other point or line meeting these conditions. Thus, in § 130 
no point not in the bisector will satisfy the conditions of the 
theorem. The center of a circle is the locus of all points in a 
plane equidistant from the circumference. 

EXERCISE 38 

1. Show that every point within an angle, and equally dis- 
tant from the sides of the angle, lies in the bisector of the angle. 

2. The bisectors of the base angles of an isosceles triangle 
form with the base an isosceles triangle. 

3. The point of intersection of the bisectors of the base 
angles of a triangle lies in the bisector of the vertical angle. 

4. Find the point in the base of a triangle w r hich is equi- 
distant from the other two sides of the triangle. Is this point 
ever the middle of the base ? 

5. Show that if lines are drawn from the middle point of 
the base of an isosceles triangle respectively perpendicular to 
the legs of the triangle, two equal triangles are formed. 

6. If from any point in the base of an isosceles triangle, 
parallels to the legs are drawn, two isosceles triangles are 
formed. Are the triangles ever equal ? 

7. Show that in a right triangle, if one angle is 30°, the 
hypotenuse is twice the shorter side. (Bisect the vertical 
angle of an equilateral triangle.) 

8. Find a point which is equidistant from two intersecting 
lines. 

9 If three lines intersect, but do not pass through the 
same point, find a point, if any such exists, which is equi- 
distant from all three lines. 

10. Bisect the exterior angles at the base of a triangle, and 
show that these bisectors meet in a point of the bisector of the 
angle at the vertex of the triangle. 



PARALLELS AND THEIR USES 119 

11. Bisect the exterior angle at the vertex of an isosceles 
triangle. Show that this bisector is parallel to the base of the 
triangle. 

12. Through the vertex of an isosceles triangle draw a 
parallel to the base. Show that this line bisects the exterior 
angle formed by extending one of the equal sides through the 
vertex. 

13. Through the middle point of one leg of an isosceles tri- 
angle draw a parallel to the other leg. Through the vertex 
draw a parallel to the base. Show that two equal triangles 
are formed. How do you find the middle point of one leg? 

14. Find a point equidistant from two parallel lines. Is 
there more than one such point? 

15. If two lines intersect, the bisectors of two adjacent angles 
formed are mutually perpendicular. 

Theorem XXII 

132. If two triangles have two sides of one equal respectively 
to tivo sides of the other, and the included angle of the first greater 
than the included angle of the second, the third side of the first is 
greater than the third side of the second. 

Draw A ABC and KLM, having sides ACQ)) and CB(a) 
respectively equal to MKQ) and ML(k), and Z C>/-M. 
We now have 

Given A ABC and KLM, with 

b = I, a = 7c. 
ZC>ZM. 
To Prove AB(c) > KL(m). 

Proof. 1. Apply A KLM to A ABC so that I will coincide 
with b, L falling at L'. Draw CO, bisecting Z L'CB, meeting 
AB at 0. Draw OV 

2. A0+ OL' > AL\ (?) 



120 MATHEMATICS 

3. In A CVO and COB, 

en = cb. (?) 

co = CO. 

ZL'CO = Z.OCB. (?) 

4. Hence, ACL'0 = ACOB, (?) 
and OX' = 05. (?) 

5. Then, ^40+ 02/ = A0+ OB. (?) 

6. Hence, A0+0B>AU, (?) 
or c>m. 

The converse of this theorem is also true. State the converse. 

Four-sided Figures 

133. A quadrilateral is a portion of a plane bounded by four 
lines. 

If the opposite sides are parallel, the figure is a parallelogram. 
Draw the figure. 

If two sides are parallel and the other two sides not parallel, 
the figure is a trapezoid. Draw a trapezoid. In an isosceles 
trapezoid the non-parallel sides are equal. 

If no two sides are parallel, the figure is a trapezium. 

A rhomboid is a parallelogram whose adjacent sides are not 
equal and whose angles are oblique. Illustrate. 

A rhombus is an equilateral parallelogram whose angles are 
oblique. 

A rectangle is a parallelogram whose angles are right angles. 

A square is an equilateral rectangle. 

The diagonal of a quadrilateral is a line drawn from one 
vertex to the opposite vertex. 

Begin lettering a four-sided figure at the lower left-hand 
corner and read counter-clockwise. 

Make a synopsis classifying quadrilaterals under three gen- 
eral heads. 



PARALLELS AND THEIR USES 121 

Theorem XXIII 

134. In a parallelogram the opposite sides are equal, and the 
opposite angles are equal. 

Draw a parallelogram ABCJD. Draw BD. Prove the two 
triangles formed are equal. Then use § 86. 

This is the fundamental theorem in parallelograms. 

EXERCISE 39 

1. The diagonal of a parallelogram divides it into two equal 
triangles. 

2. The diagonals of a parallelogram bisect each other. 

3. Parallel lines included between parallel lines are 
equal. 

4. From two opposite vertices of a parallelogram draw per- 
pendiculars to the diagonal drawn between the other vertices. 
Show that two pairs of equal triangles are formed. 

5. Show that the lines connecting the middle points of the 
opposite sides of a parallelogram bisect each other. 

6. The diagonals of a rhombus bisect each other at right 
angles. 

7. In a certain parallelogram the diagonals bisect its 
angles. One side of the parallelogram is 8'. Find the other 
sides. 

8. In an isosceles trapezoid draw perpendiculars from the 
extremities of the shorter base to the longer base. Show that 
two equal triangles are formed. 

9. The diagonals of a rectangle are equal. 

10. In an isosceles triangle ABC, the equal angles A and B 
are bisected. These bisectors form an angle of 120°. The leg 
of the triangle ABC is how many times its base? 



122 MATHEMATICS 

Theorem XXIV 

135. Two parallelograms are equal if two sides and the in- 
cluded angle of one are respectively equal to two sides and the 
included angle of the other. 

Given HI ABCD and HEFG with AD, AB, and Z A respec- 
tively equal to HG, HE, and Z H. 

To prove O ABCD = OHEFG. 

Proof. Draw diagonals DB and GE. 

Prove A ADB = A HGE. 

Then use exercise 39, 1. 

Theorem XXV 

136. If the opposite sides of a quadrilateral are equal, the 
figure is a parallelogram. 

Draw quadrilateral ABCD having AB = DC and AD = BC. 
Draw BD and prove the triangles formed are equal. 
Then use § 125. 



EXERCISE 40 

1. If two sides of a quadrilateral are equal and parallel, 
the figure is a parallelogram. 

2. If the diagonals of a quadrilateral bisect each other, the 
figure is a parallelogram. 

3. Two rectangles are equal when two adjacent sides of one 
are respectively equal to two adjacent sides of the other. 

4. The diagonals of a square bisect its angles. 

5. If the diagonals of a rectangle bisect its angles, the rec- 
tangle is equilateral. 



PARALLELS AND THEIR USES 123 

Theorem XXVI 

137. If a series of parallels intercept equal parts on one trans- 
versal, they intercept equal parts on every transversal. 

Given lis AB, CD, and EF intercepting equal parts on line 
GH. 

Ml \m' 




To prove AB, CD, and EF intercept equal parts on every 
transversal 17. 

Proof. 1. Let HG cut AB, CD, and EF at M, JST, K, respec- 
tively, and XFcut these lis at M', N\ K', respectively. Draw 
M'O and N'R respectively II to MK. 

2. MN= M'O = NK= N'R. (?) 

3. Now prove A M r ON' = A N'RK 1 . 

EXERCISE 41 

1. A line which, bisects one side of a triangle and is parallel 
to another side bisects the third side. (In § 137, if M'K' 
were drawn through M, we would have a triangle MKK' with 
J¥ the middle point on one side.) 

2. The line parallel to a base of a trapezoid, and bisecting 
one of the non-parallel sides, bisects the other non-parallel 
side. (In § 137 KK'M'M is a trapezoid.) 



124 MATHEMATICS 

Theorem XXVII 

138. The line joining the middle points of two sides of a tri- 
angle is parallel to the third side and equal to one half of it. 

Draw A ABC. Through M and M' , middle points of AB 
and AG, respectively, draw MM' . We then have 

Given A ABC with MM' connecting the middle points of 
the two sides. 

To prove MM' II BG. 

Proof. 1. Through M draw a line parallel to BG. 

2. This line will bisect AG and must therefore pass through 
M' . Why ? 

3. The parallel drawn will coincide with MM', therefore 
MM' II BG. 

4. Draw M'X II AB meeting BG at X. 

5. Prove A AMM' = A M'XC. 

6. Then, MM' = XG=i BG. 

Theorem XXVIII 

139. The line joining the middle points of the non-parallel 
sides of a trapezoid is parallel to the bases and equal to one half 
of their sum. 

Draw trapezoid ABGD, AB and DC being the parallel sides. 
Draw EF joining E and F } the middle points of AD and BG, 
respectively. We then have 

Given trapezoid ABCD, with EF joining the middle points 
of the non-parallel sides. 

To prove EF II AB and DC, also EF = i(AB + DC). 

Proof. 1. A line through E II AB will pass through F. 
(Exercise 41, 2.) 

2. Then EF II AB and DC. 

3. Draw DB, intersecting EF at M. 

4. In A ABD, EM is drawn through E the middle point of 
AD and parallel to AB. (Use exercise 41, 1.) 

5. Use A BCD in a similar manner. 



PARALLELS AND THEIR USES 125 

Theokem XXIX 

140. The bisectors of two of the angles of a triangle intersect 
on the bisector of the third angle. 

Draw A ABC, also BM bisecting Z B and CM bisecting 
Z C We now have 

Given A ABC, and BM and CM bisecting A B and C, re- 
spectively. 

To prove that M lies in the bisector of Z A. 

Proof. 1. Draw AM. 

2. Since BM bisects Z B, M is equally distant from AB 
and BC (§ 130.) 

3. Since CM bisects Z C, Mis equally distant from BC and 
CA. (§ 130.) 

4. Therefore M is equally distant from AB and AC. 

5. Then M lies in the bisector of Z A, and AM is the 
bisector. (Exercise 38, 1.) 

141. From theorem XXIX we may assume that the three 
bisectors of the angles of a triangle meet in a point, and the 
point of intersection is equally distant from the three sides. 

EXERCISE 42 

1. In a parallelogram ABCD, M, the middle point of AB, is 
joined to D, and M', the middle -point of DC, is joined to B. 
Show that the diagonal AC is trisected. 

2. Show that if the middle points of the sides of a quadri- 
lateral are joined in order, the figure formed is a parallelogram. 

3. If the middle points of the sides of a rectangle are 
joined in order, the figure formed in a rhombus. 

4. The figure formed by joining the middle points of the 
sides of a square taken in order is a square. 

5. Show that the two bisectors of the interior angles on the 
same side of the transversal of two parallel lines form a right 
angle. 

. 6. What figure is formed by joining the middle points of 
the sides of an isosceles trapezoid taken in order? 



126 MATHEMATICS 

7. Show that the opposite angles of an isosceles trapezoid 
are supplementary. 

Theorem XXX 

142. The perpendiculars erected at the middle points of two 
sides of a triangle meet in a point which lies in the perpendicular 
bisector of the third side. 

Draw the perpendicular bisectors of two of the sides. Join 
their point of intersection to the middle point of the third side, 
Use § 101. Prove in a manner similar to that used in § 140. 

143. We may conclude that the three perpendicular bisec- 
tors of the three sides of a triangle meet in a point equidistant 
from the three vertices. 

Theorem XXXI 

144. The three altitudes of a triangle meet in a common 
point. 

Draw A ABC, and its three altitudes x, y, z. We now have, 

Given A ABC and the altitudes x, y, z. 

To prove that x, y, and z have a common point. 

Proof. 1. Through A draw MK II BC\ through B, draw 
MR II AC; through C draw HK II BA. 
2 x±MK. (§121.) 

3. Since MBCA and ABCK are parallelograms, BC = MA 
= AK 

4. Hence x is the perpendicular bisector of MK. 

5. Similarly y is the perpendicular bisector of HK and z of 
MH. 

6. But the perpendicular bisectors of the sides of A MHK 
meet in a point. (143.) 

7. But x, y, z are also the altitudes of A ABC. 

8. Therefore the altitudes of a triangle meet in a common 
point. 



PARALLELS AND THEIR USES 127 

Theorem XXXII 

145. Two medians* of a triangle meet in a point of the third 
median. 

Draw A ABC, and medians CM and AK meeting at 0. Also 
draw BL through 0, meeting AC at L. We now have 
Given A ABC with medians ^4iTand CM meeting at 0. 
To prove that is in the median drawn to the third side. 

Proof. 1. Draw CH parallel to AK and meeting BL pro- 
duced at H. Draw AH. 

2. In A HBC, OK II HC and bisects BC. It therefore bisects 
BH. (?) 

3. Since is the middle point of HB, and ilfthe middle 
point of AB, MO II AH. (?) 

4. Then AOCH is a parallelogram, and L is the middle 
point of AC. Why? 

o. Hence BL is a median and lies in the median. 

146. In § 145, since H0= OB and L0 = \ HO, is § the 

distance from 5 to JO. Similarly we may show that is -| the 
distance from (7 to M, and from ^4 to X. Or the medians meet 
in a point f the distance, along the median, from the vertex to 
the opposite side. 

Polygons 

147. A polygon is a portion of a plane bounded by three or 
more straight lines. 

The bounding lines of a polygon are its sides. 

Any two adjacent sides form an angle of the polygon. 

A polygon is named with reference to its number of sides, 
or angles. Thus, triangle, quadrilateral, pentagon, hexagon, 
octagon, etc. 

* A median is a line drawn from a vertex of a triangle to the middle point 
of the opposite side. 



128 MATHEMATICS 

If the sides are equal, the polygon is equilateral. 
If the angles are equal, the polygon is equiangular. 
A regular polygon is both equiangular and equilateral. 

148. Two polygons are mutually equilateral if their corre- 
sponding sides are equal. 

Two polygons are mutually equiangular if their correspond- 
ing angles are equal, that is, if their angles taken in the same 
order are respectively equal. 

Two polygons are equal if they are both mutually equilat- 
eral and mutually equiangular. They are also equal if they 
can be separated into the same number of triangles, equal each 
to each, and similarly placed. 

Theorem XXXIII 

149. The sum of the angles of any polygon is equal to twice 
as many right angles as the polygon has sides, less four right 
angles. 

Draw polygon ABODE, etc. We have 
Given polygon ABCD, etc., having n sides. 
To prove Z. A+AB+Z C+Z.D+-- etc., =2 wrt. A— 4rt. A. 
Proof. 1. From any point P within the polygon draw a 
line to each vertex: 

2. It is evident that n triangles are formed, one for each 
side of the polygon. 

3. The sum of the angles of each triangle is two right angles, 
and the sum of the angles of all the triangles together is 2 n 
right angles. 

4. The sum of the angles of all the triangles includes the 
sum of the angles of the polygon, and the angles around P 
which do not belong to the polygon. 

5. Subtracting the sum of the angles at P from the sum of 
the angles of the n triangles, we have 2 n rt. A — 4 rt. A for the 
angles of the polygon. (§ 83, d.) 






PAKALLELS AND THEIR USES 129 

Theorem XXXIV 

150. T7ie sum of the exterior angles of a polygon formed by 
producing one side at each vertex of a polygon is equal to four 
right angles. 

Given a polygon with exterior angles b, d, f etc. 

To prove that the sum of these angles is equal to 4 rt. A. 

Proof. 1. Let a be the interior angle of the polygon adja- 
cent to b. 

2. Thena + 6 = 2rt. A. 

3. This same sum exists at each of the n vertices, therefore 
the sum of the interior and the exterior angles of the polygon 
is 2 n rt. A. 

4. But the sum of the interior is 2 n rt. A — 4 rt. A. (?) 

5. Subtracting the sum of the interior angles from the sum 
of the interior and exterior angles, we have 

2 nit. A -(2 n rt. A - 4 rt. A), 

or 4 rt. A. 

EXERCISE 43 

1. Prove Theorem XXXIII by triangles formed by drawing 
all the diagonals from any one vertex of the polygon. 

2. One angle of a regular polygon is 1^- right angles. How 
many sides has the polygon ? How many degrees is the sum 
of its angles ? 

3. One angle of a regular polygon is 120°. What kind of a 
polygon is it? 

4. An exterior angle of a regular polygon is 120°. How 
many sides has the polygon ? 

5. Each exterior angle of a polygon is 30°. Find the sum 
of the interior angles. 

6. One angle of a regular polygon is -J- rt. Z. Find the 
number of sides. Any trouble here? Why? 



130 MATHEMATICS 

7. From one vertex of a regular polygon 5 diagonals can be 
drawn. Find the sum of the angles of the polygon. 

8. What three lines belonging to any triangle meet in a 
point ? Is there more than one answer to this question ? 

9. One exterior angle of a regular polygon is 180°. How 
many sides has the polygon ? What is the greatest exterior 
angle a regular polygon can have ? What is the least exterior 
angle possible ? 

151. Develop the following synopses : 

Two lines are equal, if 

Two angles are equal, if 

Two lines are parallel, if 

A quadrilateral is a parallelogram, if 

Supplemental Applied Mathematics 

1. A hexagonal water tank is 3' — 6" on a side. Find the 
area of the cover. 

2. In nuts and heads of bolts, the distance across the flats 
(the distance between the parallel sides) is f the diameter of 
the bolt plus l". The thickness of the head is i the distance 
across the flats. Find the distance across the flats and the 
thickness of the head on a one-inch bolt having a square 
head. 

3. A hexagonal head is -§-" on a side. Find the distance 
across the flats and the thickness of the head. 

4. A square head T \" thick is to be milled on a cylindrical 
blank. Find the diameter of the blank. 

5. In example 3, find the approximate diameter of the 
bolt. 

6. A half-inch bolt has a square head. Find the distance 
across the flats, the thickness of the head and the diagonal of 
the head. 



PARALLELS AND THEIR USES 131 

7. A one-inch steel bolt is 4" long under the head. The 
head is square. Find weight of these bolts per hundred. 

8. A square head measures f " across the flats. The bolt 
is 5" long under the head. Find weight per hundred. 

9. The thickness of a hexagonal steel head is f ". Find 
weight of heads per hundred. What size bolt would you use 
with such head ? 

10. A hexagonal head is -f " on a side. Find the diameter of 
the bolt. 



11. A hole is 0.185". Could you use a y 3 g " bolt in this hole 
Would a i" bolt be too large r i 



? 



? 



12. A hole is 1.284". The longest diagonal of the hexag- 
onal head used is 2.74". Find size of bolt used, weight of nuts 
per hundred, number of nuts per hundred pounds. (Bolts 
only come in 4ths, 8ths, 16ths, 32ds, 64ths). 

13. Whites of eggs coagulate at 56.6° C. Express this tem- 
perature in Fahrenheit scale. Yolks of eggs coagulate at 
122° F. Express in Centigrade. 

14. When eggs are made into omelets, 1 tablespoonful of 
milk and ± teaspoonful of butter are added to each egg. 7 eggs 
will make 5 portions for serving. Find difference in cost of 6 
eggs and enough omelet to serve six persons, when eggs cost 
30^ per dozen, milk 6/ per quart, butter 37^ per pound. One 
cup of milk measures 16 tablespoonfuls, and one half pound of 
butter 16 teaspoonf uls. 

15. 87 °J of milk is water. 1 cup of milk weighs 8|- ounces. 
Find weight of water contained in one quart of milk. 

16. 3.3 °f of milk is protein, -f of the protein is casein and \ 
albumen. What is the per cent of casein and albumen in milk ? 

17. The edible portion of cooked eggs contains 13.2 % pro- 
tein. How much milk (liquid measure) does it take to contain 
as much protein as is in one dozen eggs ? 



132 MATHEMATICS 

18. A can of condensed milk costing 5jf contains f of, a cnp. 
Which is cheaper — fresh or condensed milk — if fresh milk 
costs 6^ per quart? (Dilute condensed milk one third.) 

19. American cheese contains 28.8 °J protein. How many 
eggs contain as much protein as 1 pound of cheese ? How 
much milk (liquid measure) in one pound of cheese? How 
much cheese contains as much protein as 2 eggs ? 

20. Find difference in cost of 1 dozen eggs and as much 
cheese as would contain the same amount of protein, cheese 
costing 20^ per pound and eggs 30^ per dozen. 

21. One quart of sour milk makes 1 cup of cottage cheese 
which weighs 6 ounces. Find difference in cost of 1 pound 
cottage and 1 pound American cheese. (Prices same as above.) 

22. Cottage cheese contains 20.9 % protein. How much 
cottage cheese contains as much protein as 1 pound American 
cheese ? How much sour milk will be required to make this 
quantity of cottage cheese ? 

23. \ pound macaroni or 1 cup rice can be used with cheese. 
1 cup rice weighs 1\ ounces, and costs 10^ per pound. Maca- 
roni costs 15 fl per pound package. Find difference in cost. 

24. Boiled rice contains 24.4 °f carbohydrates ; cooked maca- 
roni contains 15.8 °/ carbohydrates. How many quarts cooked 
macaroni will contain as much carbohydrates as 5 cups cooked 
rice ? 

25. Uncooked macaroni contains 74.1% carbohydrates; 
cooked macaroni contains 15.8 °J carbohydrates. Find weight 
of loss in carbohydrates from cooking 1 pound macaroni. 

26. I wish to make a dusting cap 18" in diameter. How 
much lace is needed to put around the edge, allowing one half 
extra for fullness ? 

27. Two inches in from the edge of the cap in Example 26 
I sew beading. How much beading must I buy ? 



CHAPTER VIII 
Products and Factors 

152. In exercises 5 and 14 we found the factors of monomial 
products. We shall now extend factoring to include the prod- 
ucts found in exercises 16 and 18. 

The Difference of Two Squares 

153. Type I. Multiply a + bby a — b. (That is, multiply 
the sum of two numbers by the difference of the same two 
numbers.) 



By actual multiplication : 



a + b 
a — b 



a 2 + ab 
-ab-b 2 



a 2 - b 2 

That is, (a + b) (a - b) = a 2 - b 2 . 

Or, stated in words : 

The product of the sum and difference of two numbers is equal 
to the difference of their squares. 

Then to multiply the sum of two numbers, as 3 x + 5, by the 
difference of the same two numbers, 3 x — 5, one needs only to 
square the first number, 3 x, and subtract the square of the 
second number, 5, from it, giving for the product, 9 x 2 — 25. 

EXERCISE 44 
Find by inspection : 

1. (x + y)(x — y). 3. (m + x)(m — x). 

2. (x + 3)(x-3). 4. (x + 4)(g - 4). 

133 



134 



MATHEMATICS 



5. (2a? + l)(2a>-l). 8. (14 y + 15) (14 y - 15). 

6. (2x + 3)(2x-3). 9. (17a + 19 6)(17a-19&). 

7. (5a + 7)(5a>-7). 10. (16c + 25d)(16c - 25 d). 

11. # 2 — 4 is the product of what two numbers ? Are these 
numbers binomials, trinomials, or monomials ? 

12. What are the factors of x 2 — 25. 

13. Eestate the rule in Type I, making it applicable for fac- 
toring such examples as example 12. Keep in mind that 
factoring is a process of division ; division by inspection. The 
dividend is given. You must find the divisor and the quotient. 

Factor the following : 

20. 16y 2 c 2 z 2 -25x 2 d 2 a\ 

21. a; 4 -225. 

22. 289 -x\ 

23. 361 x 2 - 529 y 2 . 

24. 441 y* - 729 a\ 

25. 64 a 2 - 196 b 2 . 

26. a 4 - 256 b 4 = (a 2 + 16 b 2 )(a 2 - 16 b 2 ) 

= (a 2 + 16& 2 )(a + 4&)(a-4&). 



14. 


a 2 - 25. 


15. 


a 2 - 81. 


16. 


a 2 - 36. 


17. 


25 a 2 - 36. 


18. 


16 y 2 - 25. 


19. 


16/ -25c 2 . 



27. 16 m 4 -81c 4 . 

28. (49) 2 - (25) 2 . 

29. (561) 2 - (559) 2 . 

30. (625) 2 - (576) 2 . 

35. 0/ + 21) 2 



31. 3 8 -2 8 . 

32. ( a + 3) 2 -(a-3) 2 . 

33. (x + 7) 2 - (x - 7) 2 . 

34. ( a +&) 2 -(a-6) 2 . 
(2/ - 21) 2 . 



36. (17x + 162/)(17a;-16?/) = ? 

37. (21a+23&)(21a-23&) = ? 

38. (24 m + 19 c)(24 m - 19 c) = ? 

39. (25* + 18c)(25a;-18c) = ? 



PRODUCTS AND FACTORS 135 

40. (15c + 26d)(15c- 26d) = ? 

41. T \a 2 --^b 2 . 

42. Can you factor a 2 + 25 ? Why? 

The square of the sum of two numbers. 
The square of the difference of two numbers. 

154. Type II. Multiply a + b by a + b. That is, multiply 
the sum of two numbers by the sum of the same two numbers. 

a + b 

a + b 



a 2 + ab 
+ ab + b 2 



a 2 + 2ab + b 2 

That is, (a + b) 2 =a 2 + 2ab + b 2 . 

Or, stated in words : (a is the first number and b is the second number.) 

The square of the sum of two numbers is equal to the square of 
the first number plus twice the product of the first and second, plus 
the square of the second. 

Similarly, (a -b) 2 =a 2 -2ab + b 2 . 

Or, TJie square of the difference between two numbers is equal to 
the square of the first, minus twice the product of the first and 
second, plus the square of the second. 

EXERCISE 45 
Find by inspection : 

1. (x + z)(x + z). 7. (5a + 3) 2 . 13. (9 m + 7x) 2 . 

2. (x + S)(x + 3). 8. (5a-3) 2 . 14. {IS a -15b) 2 . 

3. (a + x) 2 . 9. (5 a + 4 b) 2 . 15. (5 m- a) 2 . 

4. (a-x) 2 . 10. (6y + 5c)\ 16. (6 a -5b) 2 . 

5. (a + 2x) 2 . 11. (7x-4z) 2 . 17. (5b -6 a) 2 . 

6. (a -2 a) 2 . 12. (8c- Id) 2 . 



136 MATHEMATICS 

18. From what do you get this product : x 2 + 6 x + 9 ? 

19. From what factors do you get a 2 — 8 a + 16 ? 

20. What are the factors of x 2 - 10 x + 25 ? 

21. How can you distinguish a trinomial square ? 
Find the factors of the following : 

22. a 2 -6a + 9. 26. 9x 2 +30x + 25 9 

23. a 2 — 4 a + 4. 27. m 2 + 12m + 36. 

24. 4a 2 + 4a + l. 28. 4c 2 + 12c + 9. 

25. 9a 2 -6^ + 1. 29. 49 d 2 - 14 cd + c 2 . 

30. m 2 A: 2 + 4 mZcc + 4 c 2 . 

31. a 4 - 18 a 2 + 81 = (a 2 - 9)(a 2 - 9) = (a + 3) (a - 3) (a + 3) 
(a -3) by Type L 

32. x* - 8 x 2 + 16. 33. 16 c 4 - 72 c 2 d 2 + 81 d\ 

34. The area of a square is 25 a 2 + 4,0 ab + 16 & 2 ; find one 
side of the square. Find the dimensions when a = 5, b = — 3. 

35. One side of a square is 2 # + 5?/. What is the area? 
Find the area when x = 6 ; y = 1. 

36. Is a 2 + 8 a + 25 a square ? Why ? 

37. Is a 2 + 10 a + 24 a square ? Why ? 

38. Is a 2 + 10 a + 25 a square ? Why ? 

39. Draw a square whose side is a + 6 inches. Draw per- 
pendiculars at the ends of a and b and show that this square 
is made up of the square on line a, pins two rectangles each 
of whose areas is ab, and a square on line b. 

40. Factor a 4 — 16. Are your factors factorable? 

41. Factor x* — 81. What type are you using? 

42. (81) 2 =(80+ l) 2 

- G400 + 160 + 1 
= 6561 



PRODUCTS AND FACTORS 137 

Write the following squares by inspection: 

43. (41) 2 45. (79) 2 47. (145) 2 49. (162) 2 

44. (39) 2 46. (132) 2 48. (153) 2 50. (169) 2 

Trinomials of the form x 2 + kx + c. Binomial Factors having 
One Term Common. 

155. Type III. Multiply x + 5 by x + 3. 

x -f 5 
x +3 
x 2 + 5x 

+ 3 x + 15 
x 2 + 8x + 15 

Note that in this trinomial the first term is the square of the 
common term (x). The coefficient of x is the sum of the unlike 
terms, 5 and 3. The third term is the product of the unlike 
terms. 

Ex. Find the product of 4 x + 3 and 4 x — 9. 

By the rule: (4a: -f 3) (4 a - 9) = (4 x) 2 +(3 - 9) (4 a) -27. 

= 16x 2 -24x-27. 

EXERCISE 46 
Find by inspection: 

1. (a + 3)(a + 4). 11. (b -14) (6 + 16). 

2. ( c +.B)(c + 7). 12. (c + 115)(c-12). 

3. (2a + 5)(2a + 3). 13. (d + 16)(d + 15). 

4. ( a + 4) (a -3). 14. (2 x- 14) (2 a; + 9). 

5. ( a -4)(a + 3). 15. (2a + 14) (2 a + 9). 

6. (5a-4)(5a-4). 16. (2 a + 14) (2 x - 9). 

7. (m + 18)(m-15). 17. (y + 5)0/ -40). : 

8. (ft - 18) (ft + 16), 18. (z + 27) (z .- 5) .. 

9. (3ft + ll)(3ft-4). . 19. (ft + 28) (ft -.7-). 
10. (c + 21)(c + 22). 20. (a - 17) (a - 15). 



138 MATHEMATICS 

21. From what factors do you get x 2 + 7 x + 12 ? 

22. Factor a 2 — x — 12. 

To get the second terms of these binomials: factor the third 
term into two factors whose sum is the coefficient of the unknown 
in the second term. 

Factor the following: 

23. X 2 -2x — 8. 33. y 2 -9y-36. 

24. x 2 + 6 x + 8. 34. a 2 -16 a + 64. 

25. x 2 -6x + 8. 35. 4 a 2 + 16 a + 15. 

(2 a is the term common to each 
binomial.) 



26. x 2 + 2x — 8. 



27. a 2 + 9 a + 20. 36. 9 a 2 - 15 a — 14. 

28. a 2 - a -20. 37. 25 or + 50 x + 21. 

29. z 2 _z-72. 38. c 4 - 13 c 2 +36. 

30. c 2 + 16o + 15. 39. m 4 - 29 m 2 + 100. 

31. m 2 + 8m + 16. 40. ^ + 16^ + 64. 

32. ?/ 2 -12?/ + ll. 41. a 2 -10 a + 25. 

42. ( a + 3)(a-3)(a + 5)(a-5) = ? 

43. a 2 + 3a + 2. 45. a 2 + 2a + 2. 

44. a 2 - 3a + 2. 46. a 2 -2a + 2. 

Polynomials having a Factor Common to Each Term 
156. Type IV. Multiply 3 a + 4 b + 2 c by 5 x. 

3a + 46 + 2c 

5# 



\5ax + 20bx + 10 ex 

Each term of the product contains the common factor 5 x. This is a 
very important type of factoring. The first step to take in all examples 
is to see if the example belongs to Type IV. 

Ex. 1. Factor 4 a 2 x - 16 b 2 x. 
Each term contains the factor 4 x. 



PRODUCTS AND FACTORS 139 

Divide by 4 #. 4 x)4 a 2 x — 16 b 2 x 

a 2 -4 b 2 
The divisor is one factor, the quotient the other factor, or 

4 a 2 x - 16 b 2 x = 4 z(a 2 - 4 6 2 ). 

But, a 2 — 4 b 2 can be factored by Type I. 

Hence, 4 a 2 x - 16 b 2 x = 4 x(a + 2 5) (a - 2 5), 

This is similar to factoring such a number as 105- 
We remove a factor 3, then have 

105 = 3(35). 
Factoring 35, 

105 = 3 • 5 • 7. 

Ex. 2. Factor (2 a + b)a? - (2 a + 6)8 a? + (2 a +6)15. 

2 a + 5 )(2a + ^ 2 - (2a+5)8x+ (2 a + 6)15 
z 2 -8x + 15 

Use Type III on the quotient. Then, 

(2a + b)x 2 - (2a + &)8z+ (2 a + 5)15 = (2a + b)(x-S)(z-S). 

EXERCISE 47 
Factor the following : 

(Be sure that you cannot still further factor your result. 
Check each answer.) 

1. 3 a; 2 + 27 x. 4. a? - ab\ 

2. a 2 b + ab 2 . 5. 5^-20. 

3. 5 a + 25. 6. (a* -&>*-- (<** -&*)»*• 

7. c*(2 m + 5) - c(2 m + 5) - 12(2 m + 5). 

8. (2 c + 7)4 c 2 + (2 c + 7)20 c + (2 c + 7)21. 

9. #(# + 3) + ?/(# + 3). 10. ax + bx + ay + by. 

Factor the first two terms and the last two terms separately. 

x(a + b)+y(a + b). 

The example is now like example 9. 

11. x? — 3 x 2 — 3 x + 9. (x 2 is a factor of the first two terms 
and — 3 a factor of the last two terms.) 



140 MATHEMATICS 

12. Factor (a + b)a 2 + (a + 6)2 a& + (a + b)b\ 
Compare example 9, exercise 16. 

13. x 2 (2x-l)-x(2x-l)-12(2x-l). 
What Types did you use ? 

14. a\a 2 -b 2 )-b\a 2 -b 2 ). 

15. (2m + 5)ra 2 + (2ra + 5)8m + (2m + 5)16. 

16. (2m + 5)m 2 - (2 m + 5)17 m + (2 m + 5)16. 

17. (2m + 5)m 2 - (2 m + 5)15 m —{2m + 5)16. 

18. 25 a 2 (2 x + 5) - 20 x(2 x + 5) + 4(2 a? +5). 

19. 4ra 3 + 12m 2 + 36m. 20. 4 m 3 + 24m 2 + 36 m. 

21. Is a 2 + 9 a + 25 a square ? Why ? 

22. a 5 - 12 a 4 + 36 a 3 . 

23. (2a + l)4a 2 +(2a + l)4a + 2£+l. 

24. (a 2 -a-72)(2a 2 -18). 28. ^ + 3 ar 9 - 6a-18. 

25. (£ + 3) 2 -(£-3) 2 . 29. 8 ^+12 ^-10 a -15, 

26. (> + 3)V-(a + 3) 2 81. 30. a^ + 4^-16^-64. 

27. a 3 + 3a 2 -9a-27. 31. m 3 -4m + 7m 2 -28. 
32. 12 a 3 + 8 a 2 b - 21 ab 2 - 18 b\ 

Trinomials of the Form ax 2 + bx + c 
157. Type V. .Fmd the product of 2 x + 5 cmd 3 x + 7. 

3x + 7 



2.3.x 2 + 3.5..x+2-7.x + 5.7 
= 2 • 3 • x* + (3 • 5 + 2 • 7)x + 5 • 7 

Notice that if the first and third terms of this trinominal are combined 
by multiplication, the product 2 • 3 • 5 • 7 • x 2 , comprises all the factors 
which make up the middle term of the product. Note also that the middle 
term, 29 x, is the sum of the cross products, that is, the sum of 2x • 7 and 
3s -5. 



PRODUCTS AND FACTORS 141 

Reversing this multiplication process we may factor trinominals of this 

type. 

Ex. Factor 6x 2 + 29 a; + 35. 

1. Find the product of the first and third terms, or 210 x 2 . 

2. Factor 210 x 2 so that the sum of the factors is the middle term 29 x ; 
these factors are 14 x and 15 x. 

3. Write the trinominal in the form of a quadrinominal, using 14 x and 
15 x for the middle terms : 

6x 2 -\-29x + 35 = 6x 2 + 14 a + 15» + 35. 

4. Now use Type IV. 

2x(3x + 7)+5(3x + 7) = (3x+ 7)(2a>+ 5). 
Hence, 6 x 2 + 29x -f 35 = (3x + 7)(2x + 5). 

It does not matter whether 14 x or the 15 x is connected with 6 x 2 . 



EXERCISE 48 
Find by inspection : 

1. (2a5 + 7)(3a> + 4). 10. (10 x - 18) (7 x + 15). 

2. (5x + 2)(2x + 4:). 11. (8aj + 17)(7a-6). 

3. (6a + 5)(2a?-3). 12. (4a> + 15)(9a;-14). 

4. (5a? + 3)(3a?-4). 13. (6 a- 21) (5 a; + 18). 

5. (2 a? -7) (3 a? -4). 14. (7a + 15)(7s + 15). 

6. (2»r-7)(3aj + 4). 15. (13a?-l)(13a? + 2). 

7. (2a> + 7)(3a>-4). 16. (14 a> + 13) (13 x + 14). 

8. (5a> + 2)(2a>-4). 17. (14 x- 8) (14 x + 8). 

9. (7a> + 3)(5»-8). 18. (3 a? -75) (2 a? -4). 

Factor the following : 19. 6 x 2 + 23 x + 20. 

6 x 2 - 20 = 120x 2 . 

120 4G? = 15-05' 8 X. 

15x + 8x = 23 x. 
Then, 6 x 2 4- 23 x + 20 = 6 x 2 4- 15 x + 8 a; -f 20. 

= 3x(2x + 5) + 4(2x-f5). 
= (2x + 5)(3x + 4). 



142 MATHEMATICS 

20. 6x 2 + 22x + 20. 29. 4 a 2 -14 a; -98. 

21. 8 x 2 + 22 x + 15. 30. a 2 + 18a; + 81. 

22. 10a 2 + 19a? + 6. 31. 36x 2 + 60x + 25. 

23. 12aj*-23a> + 10. 32. 15x 2 + Ux- 16. 

24. 8^ + 2^-15. 33. 15 6 2 - 14 6 -16, 

25. 15ar> + 23a-28. 34. 15a 2 + 34a- 16. 

26. 15^ + 47^ + 88. 35. 10x 2 -x-24:. 

27. 56a 2 -17a-3. 36. 10 a 2 - 29 x + 10. 

28. 4d 2 + 12d+9. 37. 14a 2 + 53a + 14. 

38. (2a-3)8a 2 + (2a-3)22*+(2a-3)15. 

39. 8z 4 + 2ar 3 -15a 2 . 40. 60 x 3 - 115 x 2 + 50 x. 

41. (a 2 -9)4a 2 -(a 2 -9)4a + a 2 -9. 

42. 4m 2 (2m + 5)+12m(2m + 5) + 9(2m + 5). 

43. 2y 2 + y-28. 44. 5 c 3 - 11 c 2 -36c. 
45. 24m 3 + 14m 2 -3ra. 

Binomials having Both Terms Raised to the Same Power 

158. Type VI. Such types arise from the following prod- 
ucts: By actual multiplication 



That is, 



a 2 
a - 


+ ab + & 2 
-5 




a 3 


+ a 2 ft + a& 2 

- a 2 b - ah 2 - 


■b* 


a 3 
a*. 


-b\ 



a-b 



Note the form of the quotient: we shall speak of a as the 
leading letter and b as the following letter. 



PRODUCTS AND FACTORS 143 

1. The first term of the quotient contains the leading letter 
raised to a power one less than its power in the dividend. 

2. The power of the leading letter becomes less by one in 
each succeeding terra of the quotient. 

3. The following letter appears in the second term of the 
quotient and its power increases by one in each succeeding 
term. 

4. The signs of the quotient are all + . 

Similarly, g Q ~ & = ** + *** + °^ + *' 

and ^^- = a 4 + a*b + aV + ab 3 + & 4 . 



Ex. 1. p = a 2 + a.4 + (4)' 

x— 4 



Here, 
and 

Likewise, 
and 



a*- 


-b* 


a- 


-b 


a 5 - 


-& 3 


a- 


-6 


tf- 


-64 





.a; 2 + 4a; + 16. 


a is 5C, 




& is 4. 




a 3 + 6 3 _ 


= a 2 - aft + b % 


a + 6 



a + b 



Note that when the binomial divisior is all positive, the 
terms of the quotient are alternately + and — . Do not divide 
the sum of the same even powers by the sum of the roots. 

EXERCISE 49 
Factor the following: 

1. X s — f. 3. a 8 -8. 5. a 8 — 27. 

2. x 3 + f. 4. X s + 8. 6. or 5 + 27. 

7. 8 a 8 4- 27. Here a = 2a; and 6 = 3. 
8 , 8 a 8 -27. 9. 125 a 3 + 1. 



144 MATHEMATICS 

10. 125 a 3 -8. 17. a 5 + 243. 

11. 64 m 3 + 27. 18. m 3 -216. 

12. 64 m 3 -?/ 3 . 19. fr 3 + 512c 3 . 

13. 64m 3 + 27?/ 3 . 20. 125 y s - 729 d\ 

14. 8 c 3 — 216 d 3 . 21. x 6 + 27 (Regard x 6 as a cube.) 

15. ^-32. 22. x 6 + 64. 

16. a 5 +32. 23. a 6 + 216. 

Trinomials which may be referred to Type I by the addition 
and subtraction of some number. 

159. Type VII. x 4 + bx 2 + c 2 . Usually in this type one 
term is a fourth power, one is a square, and one is the product 
of a square and some number. 

Ex. Factor x 4 + 2 x 2 + 9. 

If the middle term were 6 x 2 , this trinomial would belong to Type II. 
Adding and subtracting 4x 2 , we have. 

x 4 + 2 x 2 + 9 = x 4 + 6 x 2 + 9 - 4 x 2 . 
= (x 2 + 3) 2 - (2x) 2 . 
By Type I, = (x 2 + 3 + 2 x) (x 2 + 3 - 2 x). 

EXERCISE 50 
Factor the following : 

1. ^ 4 + £ 2 + 25. 6. a 4 - 39 a 2 b 2 + 49 6 4 . 

2. 9a 4 + 8x 2 + 4. 7. a 4 + 4. 

3. a 4 + 5 a 2 + 49. 8. 4?/ 4 — 37?/- + 9. 

4. a 4 + 10 a 2 + 49. 9. 4?/ 4 + 8?/ 2 + 9. 

5. a 4 - 23 a 2 + 49. 10. 25 a 4 - 65 a 2 + 10. 

Fill in the term which will make each of the following a 
trinomial square: 

11. a 2 + 4a + ? 12. 4m 2 +28m + ? 



PRODUCTS AND FACTORS 145 

13. a; 2 — 24 a;. 19. c 4 + 16. 

14. c 2 + 4. 20. 9 m 2 — 18 m. 

15. ? + 6x + 9. 21. 25a 2 -30ab. 

16. x 2 + 8x + ? 22. 25 a 2 - 25 ab. 

17. x 2 + 5x + ? 23. 25a 2 -20a&. 

18. c 2 -9c + ? 24. 25 a 2 - 18 aft. 

160. The seven types of factorable expressions should be 
committed to memory. Before attempting to factor any ex- 
pression, select the type to which it belongs. 

Hints on Factoring 

1. For all expressions : 
First try Type IV. 

2. Test binomials by Types I, IV, VI. 

3. Test trinomials by Types II, III, V, VII. 

4. Be snre that each factor will admit of no further factor- 
ing. 

5. Several types may be needed to completely factor an 
expression, e.g., example 13, exercise 47. 

Types 
I. a *-tf = (a + b)(a-b). 

II. (a+b)(a+b) = a 2 + 2ab + b 2 . 
(a-b)(a-b) = a 2 -2ab+ b 2 . 

III. (x + a)(x+b)=x 2 + (a + b)x + ab. 

IV. ad+bd+cd=d(a + b+c). 
V. ax 2 + bx + c. 

VI. a n -b n . 
a n + b n . 
VII. x* + bx 2 + <?. 



146 MATHEMATICS 

EXERCISE 51 

Factor the following : 

1. f-2y + l. 3. y*-2y-8. 

2. y 2 -l. 4. f + 2y-8. 

5. 50x 3 + 4,5x 2 -A5x. 

6. 10x\2x-3) + 45x(2x-3)-4:5(2x-3). 

7. (341) 2 -(339) 2 . 

8. One side of a square is 585'. This square is surrounded 
by a concrete walk of uniform width, whose outside perimeter 
is 2388'. Find the cost of the walk at 14^ per square foot. 

9. One square is so placed within another that a space of 
uniform width is between the sides of the outer and the inner 
squares. The sides of the squares are 76" and 69", respectively. 
How many square feet in the difference between the areas of 
these squares? 

10. The area of a square is 36 x 2 + 84 x + 49. Find one side. 
How large is the square when x = — 4 ? If x were — 9, would 
the area still be positive ? Could you substitute a value of x 
that would make the area negative? Why? 

11. The area of a square is 49 a 2 — 28 a + 4. Find one edge 
when a = \. 

12. Is a 2 + 10 a + 16 a trinomial square ? Why ? 

13. The area of a triangle is ~~ w • Find the 

6 2 

base and altitude. If x is 2 and the triangle is isosceles, con- 
struct the triangle. Is more than one such triangle possible? 
Why? 

14. The area of a right triangle is 5 x 2 + 4 x — 12. Construct 
the triangle when x = 2. 

15. Find by inspection (2# + 3) (2a? — 3)(x + 4) (a — 4). 



PRODUCTS AND FACTORS 147 

16. Factor x 4 - 25 x 2 + 144. 22. (4a + £) 2 = ? 

17. a 4 -29 a 2 + 100 = ? 23. Factor 4 a 2 + # + 1 

18. a 4 -81 = ? 24 f 2a? + l , 4 y _ 9 



4 2^ + 1 

19. (2m + 3) 2 -36 = ? 

/g + 1 | 5 Y-9 

20. (2m-5) 2 -25m 2 = ? V 5 a + V 

21. (2 m- 5) 2 - (2 m + 5) 2 = ? 26. Factor 5 a 2 + 11 a; - 36. 

27. The area of a rectangle is a; 2 + 7 sc + 12. Find its dimen- 
sions when x = 4:". 

28. Find the dimensions of a rectangle whose area is 
4ar 2 -169 2/ 2 . 

29. Find the dimensions of a rectangle whose area is 
225 a 2 - 289 b 2 . 

30. Find the dimensions of a rectangle whose area is 

x 2 + (3 m + 4 a) a: + 12 ?7ia. 

What are its dimensions when x = 4", m = 1", a = 1" ? How 
does this rectangle compare with that in example 21 ? 

31. Compare the adjacent sides in a rectangle whose area is 

25 x 2 -4:0xy + 16y 2 . 

32. Factor 9 x* + 42 #cd + 49 c 2 d 2 . 

33. Factor 4 c 2 - 256 c 4 d 2 m 2 . 

34. 15 a 2 + 29 a -14. 

In working examples under Type V, it is not necessary to 
carry out all the steps indicated in § 157. 

(15^)(-14) = -210^ 2 , 

-210x 2 = (35x)(-6x). 

Connect either of these factors with 15 x 2 ; for example, 



148 MATHEMATICS 

15 # 2 + 35 #. 

Factor this expression, 5 x (3 x + 7). 

Then, (3 x + 7) is one factor of 15 # 2 + 29 x — 14, 5 # is the 
first term of the othei factor. The second term of the second 
factor is formed by dividing — 14 by + 7. 

The written work would appear as follows : 

Factor 15 a 2 + 29 a -14. 

(15 a 2 ) (-14) = -210 a; 2 , 

-210# 2 = (35#)(-6#), 
15# 2 + 35# = (3# + 7)(5#), 
15 # 2 + 29 #-14 = (3 # + 7) (5 #-2). 
Factor : 

35. 6y 2 + Uy-12. 42. 96 y 2 - 24 y - 12. 

36. 10 m 2 + m- 21. 43. 36 # 2 - 12 # - 120. 

37. 40c 2 + 7c-3 = 0. 44. 48s 2 + 50s + 2. 

38. 6 x 2 + #-126 = 0. 45. 31 a 2 - 151 a - 20. 

39. 10 x 2 + 51 ex + m c 2 . 46. 48 a 2 + 128 a -48. 

40. 36 x 2 - 181 # + 225 = 0. 47. 25 a 2 + 95 a -20. 

41. 30 x 2 -27 #-21. 48. 64 m 2 -40 m + 4. 

Solutions by Factoring 

161. If the product of two or more finite numbers is zero, at 
least one of the numbers is zero. 

That is, if (x — 3) (x — 4) = 0, either x — 3 or x — 4 must be 
zero. If x — 3 = 0, x = 3. If # — 4 = 0, x = 4. 

Such conditions give a method for solving equations which 
are higher than the first degree. 

Ex. Solve x 2 - 4 # = 21. 

Transposing so that the second member of the equation is 0, we have, 

£2_4 X _21=0. 



PRODUCTS AND FACTORS 149 

Factoring by Type III, 

(x-7)(x + S) = 0. 
Placing the first factor equal to zero, we have, 

x = 7. 
Placing the second factor equal to zero, 

x = -3. 

These should be the roots of the original equation. 
Checking for x = 7, 

72_4.7_21 = 0, 
49-28-21 =0. 

For x = - 3, (- 3) 2 - 4(- 3) - 21 = 0. 

9+12-21 = 0. * 
Both roots satisfy the equation. 

EXERCISE 52 
Find value : 

1. 24 (x + 5) (x — 3) (x + 1), when x = 0. 

2. 16(a? — l)(a? + 2)(a?+8), when a = 1. 

3. 1625 (x + 3) (x + 5) (x - 16), when x =— 5. 
Solve by factoring. (Check each root.) 

4. a?-7a? + 12 = 0. 

5. tf + 5y + 6 = 0. 

6. ^ + 70 + 6 = 0. 

7. a 2 - 81 = 0. 

8. ^-9 a 2 - 9^ + 81 = 0. 

9. 4.x 2 + 12 x = -9. 
10. ?/ 3 -?/ 2 - 9^ + 9 = 0. 

18. (2 a + 7) a 2 - (2 a + 7) 4 a + (2 a + 7) 4 = 0. 

19. {x 2 - 16)±x 2 -(x 2 - 16)28 x + {x 2 - 16)49 = 0. 

20. (25 z 2 - 225) (49 z 2 - 289) = 0. 

21. How many roots to a first degree equation? second 
degree? third degree? nth degree? 



11. 


v 2-8v = -16. 


12. 


x 2 -x = \2. 


13. 


m 2 — 4 m = 0. 


14. 


48^ 2 -12a; = 0. 


15. 


yi-lly-102 = 0. 


16. 


9z 2 -30z + 25 = 0. 


17. 


16z 2 + 42tf = -5. 



150 



MATHEMATICS 



Factor : 


REVIEW 


1. a 4 -25. 


28. 


2. a 2 -11a + 30. 


29. 


3. 27a 6 -a 3 o 3 . 


30. 


4. 6 4 + 49a 2 + 14a& 2 . 


31. 



37. 
38. 
39. 



5. 4a 2 -7ca 2 -4d 2 + 7cd. 32. 

6. a 2 -17 a + 72. 33. 

7. a 2 + 16 c 2 - 8 ca. 34. 

8. 2 ma — mb + 6 a — 3 b. 35. 

9. ca + 2c7i — 5 6a — 10 M. 36. 

10. 5bx — 106 + 4c?a — 8d. 

11. 24 6 2 + 37 6-72. 

12. a 4 + 15 a 2 -100. 

13. 16a 2 6 2 + 24a& + 36 6 3 . 

14. 64 + 8. Treat as a bi- 4a 
nomial. 41. 

15. 9a 2 + 9a + 2. 42. 

16. 121 2/ 4 - 81 x 2 . 43. 

17. 6a 2 - a -2. 44. 

18. b 2 + 3 b - 154. 45. 

19. A; 2 + 26 A: + 133. 46. 

20. 5 m 2 -26 m -24. 47. 

21. a 2 -19 a + 78. 48. 

22. 5 a 4 + 9 a 2 -18. 49. 

23. l-125m 3 . 50. 

24. x 2 + y 2 — 2 xy — 4. 51. 

25. 5 a?/ — 3 a 2 ?/ + 4 arty. 52. 

26. a 4 -30 a 2 ?/ + 225 ?/ 2 . 53. 

27. 4ar 2 + 16a;+ 15. 54. 



9-(x 2 -2xy + y 2 ). 

28 a; 2 + 17 a; -3. 

125 a 6 + 6 9 . 

8a 2 + 20 a£ + 4ac. 

9a 8 + 30a 4 + 25. 

a 2 -17a + 42. 

2a 2 + 21a + 55. 

8a 2 - 10a -33. 

36 + 12 + 1. Treat as a 
trinomial. 

12b 2 -476 -65. 

49 a 2 -21 a -10. 

9z 2 + 48z + 64. 

84 / + 4 2/ 3 z - 42 y z z 2 - 2 z\ 

729 v 2 - 529 w 4 . 

625z 4 -216w 4 . 

a — — - x — ^oo. 

a; 2 -2a-143. 

8a 3 -12a 2 - 10a + 15. 

121a 4 -132a 2 6 + 366 2 . 

10a 2 + 49 a -33. 

729 a 8 -841. 

2 aa 4 + 4 aa 3 — 70 aa 2 . 

169a 2 + 390a6 + 2256 2 . 

y« - 64. 

6 a 4 + 162 a. 

4a 2 + 30a + 50. 

2x 2 -5a- 25. 



PRODUCTS AND FACTORS 151 

55. 49 x i + 45 x 2 y 2 + 25 y\ 66. IS x 2 - 3 xy - 4,5 y 2 . 

56. x 2 + 6 a - 247. 67. a 4 - b\ 

57. 70^2/ -982/ 2 - 140 2/z. 68. 6 a 2 -41a- 56. 

58. x z — x 2 + a - 1. 69. 75 a 2 6 2 - 108 c 2 d\ 

59. 9a 2 + 30a + 25. 70. # 2 - 2 #?/ + ?/ 2 - z 2 . 

60. 2 an — 4 n 2 . 71. a 3 — a 2 — 9 x + 9. 

61. 4 a 3 - 6. 72. 30 a 2 - 89 a + 35. 

62 . a; 2 + a;-6. 73. 40 + 26-26 2 . 

63. x 2 -x -6. 74. 5aV-4a 2 ^- 96 z 2 . 

64. a 2 — 7 # + 6. 75. 4:X 2 — xy — 3y 2 . 

65. a 2 z 2 + 12xz — 13. 

Supplemental Applied Mathematics 

1. A 10-inch steel pipe is 10.19" inside diameter, 10.75" 
outside diameter. Find weight per lineal foot. 

2. A 9-inch pipe is 8.937" inside diameter and 9.625" out- 
side diameter. Find weight per lineal foot. 

3. A steam pipe is 12.750" outside diameter and is made of 
steel -|" thick. Find the inside diameter. 

4. The inside diameter of a 2-inch pipe is 2.067". If water 
is forced through this pipe at the rate of 10' per second, how 
many gallons can the pipe deliver per hour ? 

5. A 6-inch main, inside diameter 6.065", in the same system 
as the 2-inch main above would deliver how many gallons per 
hour? Could you use the result obtained in example 4 in 
solving this problem, and have a fairly accurate result ? 

6. A pump delivers 23.5 gallons of water per stroke and is 
set for 16 strokes per minute. What weight of water is de- 
livered per hour ? (Water weighs 621 pounds per cubic foot.) 

7. A single cylinder pump 3^" in diameter and 6" stroke 
makes 22 strokes per minute. What is the discharge in 
gallons per hour ? 



152 MATHEMATICS 

8. A technical school needs 75,000 sheets of paper 
81" x 10i". The stock is 17" x 22", 16 pounds to the ream of 
500 sheets, and costs 6f ^ per pound. How many reams must 
be ordered and what is the cost ? (This is a problem in mental 
arithmetic.) 

9. 30,000 cards 5" x 4" are to be cut from stock 17" x 20£". 
Weight 30 pounds to ream, price 12 ^ per pound. How much 
stock was bought ? What did it cost ? Does it make any dif- 
ference which way the stock is cut ? 

10. In a fuel test 100 pounds of coke was found to contain 
6.01 pounds of ash and .583 pound of sulphur. The balance 
was carbon. What per cent of carbon was there? 

11. Air is .001276 times as heavy as water. What is the 
weight of the air in your classroom ? 

12. If a body falls freely in space, the distance fallen is equal 
to ± g times the square of the time in seconds, where g is the 
force of gravity (32.15 feet). 

The equation for this law is usually written 

s = | gt 2 , where s is the distance in feet. 

A stone dropped into a canon is seen to strike the water at 
the bottom of the canon in 8 seconds. How deep is the canon ? 

13. A stone dropped over a precipice fell for 10 seconds be- 
fore striking the ground. How far did it fall ? 

14. An aeroplane was sailing 1000 feet above the ground 
when one of the officers dropped a 120 lb. bomb overboard. 
How long did it take the bomb to reach the ground ? 

15. 56 hurdles 5 feet long just reach across a field. How 
many hurdles 4 feet long would be needed ? 

16. For bronze bearings the Pennsylvania Eailroad uses the 
following alloy, 77 % copper, 8 % tin, 15 % lead. How many 
pounds of copper, tin and lead are used in making 900 pounds 
of bearings ? 



PRODUCTS AND FACTORS 153 

17. In one lot of 402 castings, 24 were spoiled, in a second 
lot of 500 castings, 38 were spoiled, in a third lot of 321, 22 
were spoiled. In which, lot was the largest percentage of loss 
due to spoiled castings ? 

18. In testing our shop drive, 9.24 horse-power went to the 
lathes and .75 horse-power went to the grindstone. The motor 
delivered 11.2 horse-power. What per cent of the power went 
to the belting and shafts ? 

19. My competitor and I handle hardware. For the same 
set of articles my competitors' prices are $ 3, $ 3.30, $ 3.55 and 
$3.70. His trade discounts are 25 %, 7|, 5 and 2. My list 
prices for the same set are $ 6.10, $ 6.70, $ 7.20 and $ 7.50 and 
my trade discount is 60 %, 7^, 7±, 5 and 2. In making a bid 
how do the net prices compare ? 

20. Three men can set up a line shaft in 8 days. How many 
men can set up the same shaft in 3 days ? * 

21. Fourteen men are at work installing the machinery in a 
shop. They work for 8 days and finish half the work. The 
work must be completed in 5 more days. How many men 
must be added ? 

22. 200 men were completing the work on the Technical 
High School. The job had to be completed Oct. 5. On Oct. 1, 
the contractor found that there were still 500 days' work to be 
done. How many men could he lay off and finish the job on 
time? 

23. One cup ground coffee makes 6 cups boiled. -| cup 
boiled coffee serves 1 person. How many level tablespoonfuls 
of ground coffee should be used for each person ? 

24. If coffee costs 35 ^ per pound and there are 4-i- cups per 
pound, find cost of enough beverage for one person. 

25. For filtered coffee -| cup is used with 5 cups water. Find 
difference in cost of enough boiled and filtered coffee to serve 
6 portions. 



154 MATHEMATICS 

26. To clear 1 quart coffee, ^ white of egg or several egg 
shells may be used. It takes 11 yolks or 9 whites of eggs to 
measure 1 cup. Assuming the yolks can be used for other 
purposes, eggs selling at 30^ per dozen, and 1 quart coffee 
used daily, how much can be saved in a month by using egg 
shells to clear the coffee ? 

27. Tea made from Ceylon tea leaves contained 8.6 % tannic 
acid after five minutes' infusion ; 10.88 °/ tannic acid after 
thirty minutes' infusion. Find the difference in the quantity of 
tannic acid extracted from tea leaves steeped for 5 and 10 
minutes during a month, if \ pint tea is used daily and 1 pint 
of the beverage weighs 1 pound. 

28. A tea contained 6.8 °J tannic acid after five minutes' 
infusion and 16.3 °J tannic acid after forty minutes' infusion. 
Find the difference in the quantity of tannic acid extracted 
from the tea leaves after steeping 5 and 40 minutes during 
a month if f pint tea is used daily. 

29. Green tea leaves contain 10.64 °J tannic acid ; black tea 
leaves contain 4.89 °/ tannic acid ; 1 cup tea leaves weighs 2 
ounces. If 1 teaspoonful tea leaves is used in making a cup of 
tea each day, find the difference in the quantities of tannic 
acid extracted from black and green tea during a month. One 
cup tea leaves measures 48 teaspoonfuls. 

30. A house-boat is 26' long, 12' wide, 10' high, and weighs 
6 tons. To what depth will it sink in the water, that is, how 
much water does it draw ? 



CHAPTER IX 

Fractions 
162. A fraction is an indicated division. It is written in the 

form -, the number above the line being the numerator or 
b' * 

dividend, the number below the line being the denominator or 

divisor. 

A fraction may be positive or negative (see Chapter III), the 
sign + indicating that the quotient is to be added, the sign — 
indicating that the quotient is to be subtracted. 

Thus, —~~ means that the quotient arising from —4 — 2 

A 

must be subtracted, the result is + 2. 

We see that three signs are involved in every fraction. The 
sign of the numerator, the sign of the denominator, and the 
sign of the fraction. Any two of these signs may be changed 
without changing the value of the fraction. 

Thus, 



+1- 


-4 -4 4 

2 ~ - 2 ~ - 2 


a- 3 


a -3 __ a + 3_ 



And — ~~ ^ 



a+6 — a — 6 — a — 6 a+6 a+6 

Such changes often simplify operations with fractions. 

Principles of Fractions 
163. The following principles govern operations in fractions : 

1. Multiplying the numerator of a fraction by a number multi- 
plies the fraction. 

155 



156 MATHEMATICS 

This depends on axiom 3, § 23, 

a x 

Multiply both sides of the equation by some number, 5, we have, 

2. Multiplying the denominator of a fraction by a number 
divides the fraction. 

The pupil may show that this principle depends on axiom 3. 

3. Multiplying both numerator and denominator of a fraction 
by the same number does not change its value. Why ? Axioms ? 

4. Dividing the numerator of a fraction by a number divides 

the fraction. 

D 

— = Q. Then - =-% . The pupil may explain use of axioms. 
d d 5 

5. Dividing the denominator multiplies the fraction. 
The pupil may illustrate. 

6. Dividing both numerator and denominator by the same 
number does not change the value of the fraction. Explain. 

Reduction 

164. Principles 3 and 6 are involved in the reduction of 
fractions, 3 in reduction to higher terms, 6 in reduction to 
lower terms. 

Ex. 1. Reduce f to higher terms. 

5 4 = 20 

6 ' 4~24* 

Ex. 2. Reduce |- to 48ths. 

6=2-3; 48 = 2 4 . 3 
2- 3 )2* . 3 

2 3 



FRACTIONS 157 

Hence, multiplying both numerator and denominator by 2 3 or 8, we have, 

5 8 = 40 

6 ' 8 48 ' 

a — 3 
Ex. 3. Reduce to a fraction whose denominator is 

a + 6 
a 2_ a _ 42 . 

a 2__ a _42=( a + 6)(a-7). 
q + 6 )(q + 6)(g-7) 
a-7 

Hence, multiplying both numerator and denominator by a — 7, we have, 

a - 3 _ (a - 3) (a - 7) _ a 2 - 10 a + 21 
a + 6 ( a + 6)(a-7) a 2 -a-42 

EXERCISE 53 

1. Reduce to 144ths : T \, |, ^g, -j^-. 

2. Reduce to 512ths : T 3 g-, ^, -g 1 ^. 

3. Reduce to 19ths : \ } \, \. 

4. Reduce to fractions whose denominator is a 2 — a — 12. 

q + 4 q-7 

a + 3* a — 4* 

2a-3 



5. Reduce to (4 a 2 + 12 x + 9)ths : 

6. Reduce to (4 a 2 — 12 a+9)ths : 



2a + 3 
2a + 3 



2x-3 

7. Reduce to (as — 5) (as + 5) (# + l)ths : 
2 Ax-1 2x+l 



a? -25' ^ + 6^ + 5' as 2 -4as-5' x + 1 

8. Reduce to (4a— l)(2aj + 5)(3#— l)ths: 

-4 3 x + 2 8^ + 5 -(a; + 2) 

8^+18^-5' 12x 2 -7x+l ] 6a; 2 +13a;-5' 12a?-7a?+l 

9. Reduce to (4 a 2 - 9) ths : 

5x + 2 2os + 4 5a? + 2 



2aj + 3' 2a; + 3' -2a + 3 



(§ 162.) 



158 MATHEMATICS 

10. Reduce to 14 (a 2 - a - 72) ths : 

8a?-l . -5 11 

165. Reduction to lower terms. 
Ex. 1. Reduce f f to lower terms. 

36 = 22 . 3 2 
54 2 • 3 3 ' 

By inspection we see that 2 • 3 2 are factors common to both numerator 
and denominator. Dividing both numerator and denominator by 2 • 3 2 
(principle 6), we have 

2 2 • 3 2 2 * 



— = 2 • 3 2 
54 



2-3- J 3 



Ex. 2. Reduce — — to lower terms. 

a 2 - 9 a + 20 

a 2 -q-12 = (q-4Xa + 3) 

a 2 -9a + 20 ( a -4)(a-5) 

Dividing both numerator and denominator by the common factor a — 4, 

we have 

(a-4)0-f 3) _ a + 3 



a 2 - 9 a + 20 



(a -4) 



(a — 4) (a — 5) a — 5 



EXERCISE 54 
Reduce to lower terms: 
1. 



3 6 1* °* 


80 
144* 


5 36 
°- 2 16* 


7 34 Q 95 
'' 102* *• 133* 


54 4. 

2 8 9* *' 


64 
5 12* 


fi 72 
b ' 2T6' 


ft _8_5_ 10 „ 54 . 
°' 119* XV " 3 2 4* 


a 2 -9 


13. 


m + 1 
m 2 + l 


^ 2 + 3^-10 
2z 2 + 3z-U 


a 2 -6a + 9 


m 2 + 2 m + 1 


14. 


m 4 — 64 


8e 3 -27d 3 

16. • 

4c 2 -9d 2 


m 2 — 1 


m 4 — 16 m 2 + 64 


4c 2 + 4cd-l, 


5d 2 


ift 


8 c 3 - 27 d 3 


8c 3 -27d 3 


12 c 3 


+ 18 c 2 d + 27 cd 2 



2. 
11. 

12. 

17. 



* The bar | as here used indicates division of both numerator and denomina- 
tor of the fraction. 



FRACTIONS 159 

19 12 c 3 + 12 c 2 d - 45 ccZ 2 2Q 9y 2 + 18yz -16z 2 

12c 3 +48c-d + 45cd 2 ' ' 18 y s + 3ti y 2 z - 32 yz 2 ' 



21. 
22. 
23. 
24. 



9y 2 -12?/z + 4z 2 
9?/ 2 -4z 2 

2Ty s -8z 3 

(18 y 2 - 12 yz)(9 y 2 z + 6 yz 2 + ±z 3 )' 

6y 2 ~13yz + 6z 2 
3y 2 + 19yz~Uz 2 ' 

(y + 7z)9 y 2 - (y + 7 z) 12 yz + Q + 7 z) 4 z 2 



3?/ 2 + 19?/z-14z 2 
(3y-2z)9y 2 + 18^z(3y-2z )- (3?/--2z)16z 2 



(27 2/ 4 -8yz 3 )(3 2/-2z) 2 



Multiples 

166. A common multiple of two or more numbers is a mul- 
tiple of each of them, e.g., 48 is a common multiple of 12 and 
16. A common multiple must contain all the factors of the 
numbers involved. 

Ex. 1. Find a common multiple of 18 and 24. 

18 = 2 • 3 2 . 
24 = 2 3 • 3. 

The different factors concerned are 2 and 3. 

The common multiple of these numbers must be made up of the prod- 
uct of 2 at least three times as a factor and 3 at least twice as a factor. 
That is, a number cannot contain 18 an integral number of times unless 
it has as factors 2 • 3 2 . It cannot contain 24 an integral number of times 
unless it has as factors 2 3 • 3. To contain both 18 and 24 it must at least 
have the factors 2 3 ■ 3 2 . The common multiples of 18 and 24 are, therefore, 

23. 3 2 , 2 4 -3 2 , 2 3 -3 3 , 2*-3 3 , 2 3 . 3 2 ■ 5, etc., 
or 72, 144, 216, 312, 360, etc. 

167. The lowest common multiple contains all the factors of 
the numbers involved the least number of times. 



160 MATHEMATICS 

To find the 1. c. m., form a product of all the different factors 
of the given numbers, and give to each factor the highest expo- 
nent found in any of the numbers. 

Ex. 1. Find the 1. c. m. of 45, 48, 54. 

45 = 3 2 • 5. 
54 = 2 • 3 3 . 

48 = 2* . 3. 

The 1. c. m. is, therefore, 2 4 . 3 2 . 5. 

Ex. 2. Find the 1. c. m. of a 2 - 9, a 2 - 6 a + 9, a 2 + 3 a - 18. 

a 2 -9=(a+3)(a-3). 
a 2_6a + 9 = (a- 3) 2 . 
a 2 + 3 a - 18 = (a - 3)(a + 6). 
The 1. c. m. is (a - 3) 2 (a + 3) (a + 6). 

How many times will this 1. c. m. contain a 2 -f 3 a — 18 ? 
Ex. 3. Find the 1. c. m. of x - 1, x 2 + 9, a - 5. 
These numbers are all prime, their 1. c.m. is their product, or 
(x-l)(x 2 + 9)0-5). 

EXERCISE 55 
Find 1. c. m. : 

1. n s + 2 7i 2 , n s -±n. 3. 72, 48, 27. 

2. a 2 -9, a 2 - a- 12. 4. 120, 18, 30. 

5. 20, a 2 - 16, 3a 2 - 6 a -24. 

6. 15, 3 x 2 -13 x -10, a 2 -25, 12. 

7. 4a 2 -l, 12x 2 + 12a + 3, 4a 2 -4a + l. 

8. z 3 -27, z 2 + 3z + 9, z 2 _9. 

9. m 2 + 3m-10, 2m 2 + 7m-15, 2m 2 -7m + 6. 
10. x + 3, a — 3, x 2 — 6. 

Addition » 

168. Fractions, like other numbers, must be of the same de- 
nomination before they can be added. To reduce fractions to 



FRACTIONS 161 

the same denomination, the lowest common multiple of their 
denominators must be found, and principle 3 (§ 163) employed. 

Ex. 1. Add / ¥ , ft, T \. 

By § 167, the 1. c. m. of 24, 18, 16 is 2* . 3 2 . 

In such work as reducing to 1. c. d. , always divide by factors. 



and 



2* • 3 2 - 24 = 2 • ?. 




Multiplying 7 and 24 by 2 • 3, we have 




7 7-2-3 42 




24 24 • 2 - 3 144 




Similarly, 5 = 5 ' 2 * = 40 , 
J ' 18 18- 2 3 144' 




d 3 __ 3 - 3 2 __ 27 




16 16 - 3 2 144 




Then, ! + A + A = i? iO _27 ; 
24 18 16 144 144 144 


_109 
144 



Ex. 2. Find the sum of 

a + 2 a 2a — 1 



a 2 -25 a 2 -10 a + 25 3 (a 2 -2a -15) 

a 2 _ 25 = (a + 5) (a - 5). 
a 2 -10a + 25= (a - 5) 2 . 
3 (a 2 - 2 a - 15) = 3 (a - 5) (a + 3). 

1. c. m. = 3 (a + 5) (a + 3) (a - 6) 2 . 
3 (a + 5) (a + 3) (a - 5) 2 ~ (a 2 - 25) = 3 (a + 3) (a - 5). 

Then a+2 = 3 (a + 2) (a + 3) (a - 5) 

a 2 -25 3 (a + 5) (a + 3) (a-5) 2 ' 

3 (a + 5) (a + 3) (a - 5) 2 - (a 2 - 10 a + 25) = 3 (a + 5) (a + 3), 

and g = 3a(q + 5)(a + 3) 

a 2 - 10 a + 25 3 (a + 5) (a + 3 ) (a - 5) 2 ' 

and 2g-l = (2.g-l)(q + 5)(g-5) 

3 (a 2 - 2a - 15) 3 (a + 5) (a + 3) (a - 5) 2 ' 

Hence, a + 2 + a 2a - 1 



a 2_25 a 2_ 10a + 25 3(a 2 -2a-15) 



162 MATHEMATICS 

= 3 (a + 2) (a + 3) (a - 5) 3g (g + 5)(g + 3) 

3 (a + 5) (a + 3) (a - 5) 2 3 (a + 5) (a + 3) (a - 5) 2 

_ (2a-l)(a + 5) (a - 5) 
3 (a + 5) (a + 3) (a - 5) 2 

^ 3 a 3 - 57 a - 90 + 3 a 3 + 24 a 2 + 45 a - 2 a 3 + a 2 ,+ 50 a - 25 

3 (a + 5) (a + 3) (a - 5) 2 

(Note the change of sign in the last numerator. Why ? Always watch 
for such negative numerators). 

_ 4 a 3 + 25 a 2 + 38 a -115 

3 (a + 5) ( a +3)(a-5) 2 ' 

Ex. 3. Find the sum of — — -\ — . 

x — 2 x — 3 x — 4 

1. c. m. = (x - 2) (x - 3) (x - 4). 
Then, -i ?- + 



x — 2 cc — 3 x — 4 

= 1 (^ - 3) (x - 4) - 2 (g - 2) (a - 4) + 3 (x - 2) (x- 3) 
~ 0-2) (or— 3) (x - 4) 

= x 2 - 7x + 12-2x 2 + 12x- 16 + 3x 2 - 15x + 6 
~ (x-2) (z_3)(x-4) 

_ 2 y 2 - 10 x + 2 

(z - 2) (x - 3) (x - 4) ' 

EXERCISE 56 
Find the following sums : 

1,1 a , 6 



# +- 2 # + 5 a + 6 a — 6 

11 fi c+1 c-1 

O. 



x+2 x+5 c — 4 c+-4 

# — 1 #+1 9 8 

tf-4: x + 2' ' c 2 -25 c 2 + 5c' 

- a; + 2 # + 3 m . m , 

4. — ! 1 -I 8. 1 + m. 

# 2 — 9 x 2 — 6 x + 9 m + 1 m — 1 

9. _J 4 5 5 



a; 2 -a;-12 a;*-16 ^ + 7 a; + 12 



FRACTIONS 163 

10 JL J— ii 2x ~ 1 2x-5 

4# — 4 6# + 6 ' x — 2 # — 4 

V x , xy 

12. — 2 1 2 — . 

x — y x + y x 2 -\- y 2 

13. _^ + -^ + 



1 + x 1 — a? # 2 — 1 

14. _«_ + J=*._l. 

a — 6 6 + a 



15. 2a ! 



a 2 — b 2 a -\- b b — a 

, _ x . 3 x 6 x 2 
16. - — ■ f- 



1 + # x — 1 1 — # 2 
a; 3 # 6 a? 2 



I + # # — 1 1 — x 2 

18 3 + 2y 16i/-y 2 2-3y 
2-y y 2 -± 2 + y 

19. _°L+&_^. 20. l + ( a - 6 > 2 . 

6 — a a + 6 (a + b) 2 

21 i (a- ft) 2 , 22 g + & g " h 

(a + b) 2 ' ' a—b a+b 

23 5 ( x + y) 5(a?-y) 

x 2 + 4 spy + 3 y 2 x 2 + 2xy-3y 2 

1 1.6 

24. 



25. 



2 a + 3 2a-3 4a 2 + 9 

2.^ + 1 4 + 5a? t 
2a:-l l-2a/ 



26. 2a? + 1 - 4 + a? +3. 
2 a - 1 1-4 x 2 



164 MATHEMATICS 

27 x2 + 1 i 4fl? = ^ 3 + 3 x * + g + 3 + 4 cc 2 - 4 x 
x*--l m + 3 (aj2_i)(aj + 3) 

^ + 7 a j2_ 3a ._ h 3 



^3 + 3o;2_^_3 

If the degree of the numerator is equal to or greater than that of the 
denominator, the fraction is improper, and may be reduced, as in arithme- 
tic, by dividing the numerator by the denominator. Then 



x 3 + 3 x 2 -x - 3 x s + 3 x 2 - x - 3 



Always reduce the result to its simplest form 
28. 1 + ^+ * . 

X — - 1 # 2 — 1 



29 a 2 +l , a-3 , 
* a» + 5 a+-2 

30 a + b + a ~fr . 
a — b a + b 

By — 3 c 5y + 3c 
25 ?/ 2 - 9 5^-3 * 

32 5y-3c . 5^/ + 3c 
25?/ 2 -9c 2 5y-3c 



33. 



2 c 2 + 7 c - 4 4 & + 8 c - 5 
2c 2 + Hc-6 2c ? + 5c-3 



34 3 m 2 + 13 m - 10 3 m 2 + 17 m + 10 
3 m 2 + 17 m + 10 3 m 2 + 13 m - 10 

a 2 - a- 56 a 2 - 25 



35. 



a 2 + 14a + 49 a 2 -2a-35 



„ 2a + 2b t 2a-2b t Sab 



37. 
38. 



a — b a + b a 2 —b 2 

a + b a—b 4 ab 
a — b a + b a 2 — b 2 

3c + 5 9c 2 + 25 
3 C __5 9c 2 -25* 



FRACTIONS 165 



39 a-2 x + 2 g?-4 
jb + 2 *-2 a; 2 + 4 

a;— 3 a; + 5 , a? — 7 



a; 2 -2a:-35 a; 2 -10a: + 21 a; 2 + 2a:-15 
41. -i ^+ c2 



4 c 3 -8 
Express as a single fraction : 

42. 4+^>. 43. 2 « 2 -9 2> 

a: + 5 a 2 + 9 

44 a2 + 9 1 

a 2 -9 

2 . 2 



45. 2 + 



47. 


41 _ 3 _i 1 
204 1 25 ' 51' 


49. 


5 x ,14 X 
76 + 95 X - 



a? 2 — 25 5a 2 -125 

4R 11_1_17_85 
™" 2~8"8 • 3 6 4 8* 

1 1 , a 

48. h - • 

75 81 a; 

169. To reduce a fraction to a whole or mixed number 
principle 6 must be employed. We choose the denominator 
of the fraction as the number by which the numerator and 
denominator must be divided. 

K /v*J /y>2 I A /v» R 

Ex. Reduce — to a whole or mixed number. 

Dividing both numerator and denominator by 2 x 2 — 3 x + 7, 
we have ^z + ^- 15 * + 115 



2 4 4(2z 2 -3z+7) 
This result is a mixed number, and the first two terms are integral. 

170. An algebraic expression is integral if its denominator 
is numerical. That is, if its denominator is 1, 2, 3, 4, etc. 

In the result of the example in § 169, f a? and \ 3 - are in- 
tegral algebraic expressions, though not integral arithmetic 
expressions. 



166 MATHEMATICS 

EXERCISE 57 

Reduce to whole or mixed numbers : 
., 4.^ + 1 * x s + 5 x 2 - 7 x + 9 

8^+64y 3 

2a + 4?/ 

a; 4 + 81 



5. 



10 



4 


x-1 


X 2 


+ 5x + 6 




x + 2 


X 2 


+ 5x + 6 




x — 3 


7 


v?-hx + VZ 




3x + l 


x s 


+ 3a? + 3x + l 




rf _|_ 2 x + 1 


X 


-1 x-2 



8. 



# + 3 



9 g + 1 , s + 2 g^ + 3 

# + 2 # + 3 £ + 4 



(Reduce each fraction sepa- 
rately, then add the results.) 



x— 2 #— 3 
(Reduce to mixed numbers, then combine.) 



1]L a? 4- 4 # + 5 a? + 2 a; + 3 2# + 5 t 
a: + 5 a; + 6 a; + 3 a; + 4 a? + 2 


Multiplication 


171. Multiplication of fractions involves principles 1 and 6 


Ex. 1. Multiply if by 5. 


— • 5 = — — • (Never use cancellation.) 
25 25 


18-5_ 5 


18- 5 _ 18 


25 


25 5 


Ex. 2. Multiply ff by 3. 


18 3 _18.3_54 


25 25 25 


Ex. 3. Multiply - + - by a + 4. 
ct — a — iZ 


a+2 .(a + 4) = - 

(a-4)(a + 3) v 


[a + 2) (a + 4) _ a 2 + 6 a -f 8 
(a-4)(a + 3) a 2 - a- 12* 



FRACTIONS 167 

Ex. 4. Multiply 2 U + 2 , by a - 4 

_« + *- . (a - 4) = (" + *)(«-*) = ^±2 . (Principle 6 .) 
Ex.5. Multiply f by T %. 

8 8 

But our multiplier is y 1 - of 4. Hence our product is 

— + 15 or ^^ = -. (Principles 2 and 6.) 
8 8 • 15 6 



EXERCISE 58 
Simplify, using factoring: 

1. — -28. 2. — - 51. 



4. 



196 289 361 



_ m 2 + 6 m + 8 , , , K N 
6. — - 1 ! — • (wr + om). 

14.4. nn V + 86 _166 2 -8& + l 



28 ' 2D' 10. 



8. 



46-1 3 6 + 24 

(a + 2) 2 (a 2 — 5a + 6) 



11. 



16 6 
c 2 ~ 25 "" t^o " 125 ac ' 28 ab ' 



c + 5 c 2 - 10 c 4- 16 5 a 3 98 6 3 75 c< 

12. 



13. 



14. 



15. 



m 2 + 5 m + 6 m 2 + 7 m 4- 10 m + 1 

m + 5 m 2 + 4m + 4 m 2 + 4 m + 3 

1Q c 2_21 c _iq 3c 2 -8c-3 m 4c 2 + llc + 6 
12c 2 + 13c + 3 2c 2 -c-10 " 5c 2 -13c-6' 

' c« + 7c + 6 \A 12c + 17 
c 2 — 3 c — 10A c 2 + 8c + 12 



168 MATHEMATICS 

18 a; 2 + a?y + ?/ 2 . a? + 9?/ 

x 2 - 81 2/ 2 (a? - ?/)(^ 2 -fay + 2/ 2 ) 



/ , 2 +^+iy, + 

f x + 3 x + 5 \ A a; 2 + 8 x + I 
\x + 5 x + 3J\ a; 2 + 6 a; + 9 

x + 5 , « + 3 A , ^ 2 + 8 a? 4- 15 \ 



19. 



21. 



Division 
172. Division depends on principles 2 and 6. 



Ex.1, f-v-3. 




9 . 3 _ 9 32 _ 3 
5 5.3 5.3 5 




T 2 — 1 

Ex. 2. Divide x o ± by a; - 1. 

a; 2 -4 J 




* 2 -l . , n _ (s + l)(*-l) 


_ x + 1 


x 2 -4'^ "^ ( x + 2)(x-2)(x-l) 


£ 2 — 4 


Ex. 3. Divide f by ||. 




1:21= 7 • 
8 8-21 





But our divisor is T ^th of 21, heuce our quotient is 12 times as large as 
though the divisor were 21. Hence, 



I _gW _±_\ 12 or I^]j^ 7 -2 2 .3 = 1 
8 " 12 \8 • 21^ 8-21 2 3 -3. 7 2 



8 

1 7 • 2 2 • 3 

What divisor was used to produce - from „ ? 

F 2 2 3 • 3 • 7 

Therefore, to divide a fraction by a fraction invert the terms 
of the divisor and proceed as in multiplication. 



FRACTIONS 169 



EXERCISE 59 
Simplify the following : 

1 14 . 6 -288(a-& 2 ) . 36(a-fr 2 ) 2 
" 15 " ' ' 289(a 2 -6) * 51 (a 2 - 6) 2 

2 102_ i _._ 17 . a 2_25 . a 2 -8a-f 15 
105 ' ^ j " a 2 - 9 ' a 2 - 6 a + 9 ' 



-1 A tr I •_> A / ~2 K / 



102 . /- 17\ 8 a 2 - 17 a + 16 

105 : V 30 y " a 2 - 5 

50 a 2 6 _^ 15 a& 2 a — b _^a + b 

33 c 3 "" 44 c 4 " ' aT&^a^' 

5 169(^ + ,V) 2 . 13(a?+y) 1Q z 3 -27 . & + 3z + 9 

lU(x-yf ' l^{x-yf ' z 2 + 9 ' Sz s + 27z 

10c 2 + 17c + 3 . 2c 2 + 17c + 21 



11. 



2c 2 + 13c-7 10c 2 -3c-l 



^ + 10^ + 21 „ 8az 

12. ' ^ TTTT- 13. 



a 2 + 12 a: + 36 (a? + 2 a 2 ) 

or 7 + 3 x - 28 4 aa; 

11 11 



15 * y_ 16 x + y x-y 

+ 



1 


1 




X 


2/ 




X 


-0 


a + y 




2/ 


# 


X 


-» + 


£ + ?/ 



17. — 18. 



c b + c 



__b b-c 
x y c 

19. Divide b - cby c-b. 20. -f-f + x + 2y 

ar + y 2 x — 2y 

* Division written in this form is called a complex fraction. 



170 MATHEMATICS 



l^jx-yf 17 2 (x- ? /) 2 



22. 



a 2 . . b 2 a ' b 

b A or o a 



b_ _a ^__9 4-^ 

aba 2 b 2 



23. Transform 24. Transform 

a — b • a ^c + a; . # 



into -. o — — mt0 - 

a + b ± b 2c-x 2c_ 1 

a x 

Fractional Equations 

173. Equations involving fractions are solved by first reduc- 
ing to integral equations by means of principles of § 167, and 
Axiom 3, then solving as in exercise 8. 

Ex.1. Solve l£±I = 5--. 

X X 

Thel. c. m. is x (§ 167). 

Multiplying both sides of the equation by x (Ax. 3), 

'2x 



z j \ x) 



X 

We have, 2 x + 7 = 5 x — 2, 

or — 3 x = — 9, 

x = S. (Ax. 4.) 

Check this root by substituting in the given or original equation. 

2 - 3 + 7 = 5-g, or ^ = 5-1. 

3 3' 3 

Ex. 2. Solve ^— ^ = ^— ? - i. 

a-2 a?-3 2 

1. c. m. =(a-2)(se-3)2. 

Multiplying by 1. c. m. 

2x 2 -8x+6 = 2^-8x + 8-x 2 + 5x-6. 
Transposing, x 2 — 5 a; + 4 = 0. 

Factoring by Type III, (a - 4) (x - 1) = 0. 
Solving by § 161, X = 4 or 1. 



Check for x = 4. 



or 



For x = 1, 



FRACTIONS 171 

4-1 ^ 4-2 1 
4_2 4-3 2' 

2 12' 
1_1 l_2 1 



1-2 1-3 2 



or = 



Ex.3. 



2 2 

3 a- 6 



a>-5 £ + 4 # 2 -a;-20 
1. c. m. = (x— 5)(x-}-4). 
Multiplying by 1. c. m. 

x + 4: + x-5 = 3x-6. 

Transposing, — x = — 5. 



Check 



x = 5. 
1 1 3-5-6 



5_5 5 + 4 52_5_20 



1,1 9 

or - + - = _. 

9 

But it is not allowable to divide by zero. 

Hence 5 is not a root. The equation has no solution. It is not an 
equation of condition. It is simply a statement that two numbers are 
equal, § 14, but the statement is not true. 

EXERCISE 60 

Solve and verify the following: 

5. ^ + l=:3x + ±. 

„ 1 1 3^+2 



1. 


V-a 


2. 


A =8 

2x 


3. 


5 _16 




X 


4. 


6 = 11. 



X 



x-1 x-2 x 2 -3x+2 

1 1 5x- 9 

" a-1 a-2 a 2 -3a+2' 

2^,3^ + 4 6x-15 

8. ! — = . 

5 2a-l 15 

(Unite the first and third fractions be- 
fore clearing the equation of fractions.) 



172 MATHEMATICS 

l 18 . 10 x 6x — 3.5a? 

9. 1 = Li X* 

7 21 5 3 

in x + 5 1= 2a; + 4 
' 4 9 

6 11 v ' 3 

., 5i 1 8/ 5\ , 7 A 
12. f oj — + — = 0. 

6 2 8^ 3) 32 
, 3a; a; + l 2x — 1 , 1 

JLo. = • 

2 3 3 4 

14 JL _^ = -A. 
2x 24 3aT 

nK 2# + 4 Q l a-3 , x + 2 

3 3 4 3 

16 - — a; + 3 = 10 ~ x _ 2 
'25 4 

x + 3 _ x — 2 _ 1 _ 3 a; — 5 
' 2 3 4~ 12 

18. 4 *- 2 + 4 - 3a; - 5 = 5. 
11 13 

._ 5-3a; , 3-5a;_3 5a; 

4 3 2 3 

20. 3 + 4a; + 3a; + 2i = 0. 

4 ' •* 

21. i + 4 n- T = T + 26. 

22. 4r l - 3 JL±2 = 3 JL±9 + 5 . 

3 4 

23. 3 £ZLl + 3 = ^i + 3^ ± 5_ 2i 

3 6 4 2 



FRACTIONS 173 



24. 3- 5 - 2x =i- A - 7x + ^+l 
5 10 2 

__ 17 a — 5 10a + 2_5a + 7 _ 

2o. — — o. 



26. g £ + 3 + 5(2 a! + 10) = 2a! + 24 

2 5 

27. 5(2 a? + 10) = 2a , 5^ + 3 , 

5 2 



28. ^ + ?! + « + JL = 82. 
2 3 4 10 

2 9 . 13 6 + 7 -^+8 = 25 + 9 + 38> 

7 o 

30 5a + 7 —2a — 4 _ 3a + 9 ~ 

2 3 4 

on a-1 11 -13 a x-2 



2 12 3 

oo 7 1 3 1 
32. = 1. 

10 4 y 5y 

33. l3 a ._ 8 ^ + Lf^-l5 a; + 22. 

9 2 

34. 2 _£_ 3 _2 = ^_32. 

5 2 2 

35. ^ + 31-^ = 9. 

5 7 2 

2a: 3a; + 4 _ 6 a; + 5 
' 5 2<c-l" 15 

37. ? + ^ = 4a;-2a. Solve for x. 
2 3 

38. - + - = ? + ?. Solve for a;, 
a; a a; a 



174 MATHEMATICS 

39 4y-l 2y + 7 ^ $y + 7 
6 4 y -8 12 

40. L±^ = lL Solve fort 
£-a 4 

t + a t — a t — 3a 



41. 



£ — a £ + a t 2 — a? 



6a? + 14s + 8 == 4(3a ? -5)(s + 4) 

3 6 . ' 

. a? x a — b 

TCO. 



44. 



45. 



a + 6 a — 6 (a + 6) 2 
15 * 2 -f 5 2 25 3 2i 



9 z 2 - 25 5 + 3 z 5 - 3 z 

3 m 2 — 2 m + 1 3m 2 + m — 2 = — 19 m 2 + 3 m + 1 
m + 3 m — 3 m 2 — 9 



46. — — 1 = 2. Solve for m. 

m — a 2m + a 

47 4 * + l __ 2z + 3 _ 3z-5 = 12z-4 
3 5 10 15 

M „ 3 fc + a 5&— a,2ft + 3a 7 A; + 2 a al o , 

48. — 1 — = — Solve for k. 

2a 4a 6a 10 a 

Ae% 5z + 2d 2 2z + 8d 2 15 z - d 2 al « 

49. ! = — • Solve for z. 

Id 2 z + 9d 2 21 d 2 

50. -JL + i - «ir-26 



^_5 ^-6 y 2 -lly + 30 
1,1 5m -23 

51. -J- 



m — 5 m — 6 m 2 — 11 m + 30 

52 6k 27c = 8 ( fe2 ~ 4 ) 

*A; + 5 2-fc ft 2 + 3ft -10* 



FRACTIONS 175 



53. 



3-2 z - 4 z-3 



(Combine the first two fractions, then the second two fractions before 
clearing the equation of fractions. ) 

54. c y + m y - y= -J— + ±. 

c c — m m 



55. 



d-3 d-± _ d-l d-2 
d-± d-5~d-2 d-3 



_. #+1 a-2 2x + 8 
ob. j- = • 

x + 3 x — 4 x + 4 

57. There is a number such that one fifth of it is greater by 
3 than one sixth of it. What is the number ? 

58. The denominator of a fraction is 5 more than the nu- 
merator. If 9 is added to the numerator and 7 subtracted from 
the denominator, the result is 1±±. Find the fraction. 

59. Find three consecutive numbers which will satisfy these 
conditions : if the smallest is multiplied by six and three sub- 
tracted from the product, this product divided by the greatest 
number will give a quotient of five. 

60. I have two proper fractions. The denominator of each 
is one more than its numerator, the numerator of the first, the 
numerator of the second, and the denominator of the second 
are consecutive numbers, and when the greater fraction is sub- 
tracted from the lesser the quotient is — £%. Find the frac- 
tions. 

61. Divide 48 into two parts, such that the fraction formed 
by these parts is ^. 

62. Two Ohio cities are 120 miles apart. Two trains run- 
ning between these cities have a difference in rate of 5 miles 
per hour, and the difference in time it takes them to make the 
run is 20 minutes. Find the rate of each train. 

63. Divide 48 into two such parts that ^ the first part 
plus -| the second is 20. 



176 MATHEMATICS 

64. From a tank one half full of crude oil 1000 gallons are 
drawn out and 75 gallons are lost by evaporation and leakage. 
The tank is then one third full. How much does the tank hold 
when full ? • 

65. A machinist and his helper receive together $ 42.80 for 
a certain piece of work. If the machinist is worth 2\ times as 
much as his helper, how much does each receive ? 

66. The width of a rectangle is f of its length. The perim- 
eter is 216 feet. Find the area of the rectangle. 

67. One fourth of a certain number plus one twelfth of that 
number equals 16. Find the number. 

68. If 42 is the sum of two numbers, and \ of their differ- 
ence is i-, what are the numbers ? 

69. The sum of the angles of a triangle is 180°. Find the 
angles of a triangle if the first is 25° more than the second, 
and the third is three times the first. 

70. Find the angles of a triangle if the first angle is double 
the second, and the third is 9° less than three times the first. 

71. A boy spent one fourth of his money, and then received 
$ 2. He spent one half of what he then had, and found he had 
$ 7 remaining. How much had he at first ? 

72. Three sons were left a legacy, of which the eldest re- 
ceived | , the second -i, and the third the remainder, which was 
$ 200. How much did each receive ? 

73. A, B and C own 10,000 head of cattle. B owns three 
times as many as A, and C owns \ as many as are owned by 
A and B together. How many does each own ? 

74. If you were earning a salary, and spent \ of it for board, 
and i of the rest for other expenses, and saved annually, $ 280, 
how much were your earnings ? 

75. The sum of the angles of a triangle is 180°. Find each 
angle if the second angle is twice the first and the third angle 
is 30° more than the second. 



FRACTIONS 177 

76. A local train loses six of its passengers at the first stop, 
one third of the remainder at the second stop, one half of the 
remainder at the third stop, and the 30 who then remain ride 
to the end of the line. How many passengers on the train 
when it started ? 

77. In example 74, the first station is -^ as far out as the 
second, the third is 36 miles beyond the second, the fourth 49 
miles beyond the third, and the length of the run is 117 miles. 
At 2 $ per mile per passenger, how much did the railway com- 
pany receive ? 

78. A number exceeds the sum of its one third, one fourth, 
and one fifth by 13. What is the number ? 

79. An oil tank can be filled by one pump in 8 hours, and 
by a second pump in 14 hours. How long does it take to fill 
the tank when both pumps are working ? 

Let x = the time when both pumps are working. 

Then - is the amount of work both pumps do in 1 hour. 
x 

- is the amount of work the first pump does in 1 hour, and 
8 

— is the amount the second pump does in 1 hour. 
14 

Then - + — = 1 Solve. 
8 14 x 

80. An oil tank 20' in diameter can be filled by one pump 
in 177^ hours, and by a second pump in 64 hours. How long 
does it take to fill the tank when both pumps are working ? 

* 81. In example 78, the cylinder of the larger pump is 10" 
in diameter, the stroke of the piston is 12", and there are 15 
strokes of the pump per minute. What is the capacity of the 
tank in cubic feet ? In gallons ? 

82. Separate 45 into two such parts that one part divided by 
the other will give a quotient of 2 and a remainder of 9. 

83. Separate 45 into two such parts that one part divided by 
the other will give a quotient of 5 and a remainder of 37. 



178 MATHEMATICS 

84. Separate 45 into two such parts that one part divided by 
the other will give a quotient of 5 and a remainder of 48. 

85. Two men, 58 miles apart, start at the same time to travel 
toward each other. The first travels 7 miles in 2 hours, the 
second travels 15 miles in 4 hours. How far does the first one 
travel before they meet ? 

86. The denominator of a fraction is 5 less than the numera- 
tor. If 5 is added to the numerator, the value of the fraction 
is then |. Find the fraction. 

87. Two men, A and B, 58 miles apart, start at the same 
time to travel toward each other. A travels 7 miles in 2 hours 
and B travels 15 miles in 4 hours. A meets with an accident 
and is delayed 20 minutes. How far does B travel before they 
meet? 

88. Separate a into two parts such that one part divided by 
the other will give a quotient of g and a remainder of c. 

89. Separate 42 into three parts such that the second shall 
be four times the first and the third 4 times the second. 

90. Separate c into three parts such that the second shall be 
m times the first and the third m times the second. 

91. A has $ 1200 out at interest, part of 6 % and the balance 
at 5 % . The part at 6 % brings as much interest in 4 years as 
the part at 5% brings in 6 years. What is his total amount 
of interest per year ? 

92 4(58-^) ^2^ 
15 7 ' 

Multiplying by 105, 

28 • 58 - 28 x = 30 x. 

58 x = 28 • 58. 

x = 28. 

(Indicate such products and save computation.) 

q , 2i-x_ q ,5x 5*-2 7<c-6 _ 15a;-5 

93 - "l6~~ + 48' 6 +5Z + 13 - 18 ' 



FRACTIONS 179 

REVIEW 

In solving the following equations, use two unknown quan- 
tities wherever possible, § 50 : 

1. A father's age now is 4 times as great as that of his son ; 
and 4 years ago it was six times as great. What are their ages ? 

2. $ 6 is changed into 51 coins. If each coin is either a 
quarter or a dime, how many of each are there ? 

3. A train leaves a station and travels at the rate of 40 
miles an hour. Two hours later a second train leaves the 
station and travels in the same direction at the rate of 55 miles 
an hour. Where will the second train pass the first ? 

4. A room is 2 feet longer than it is wide, and if its length 
were increased by 4 feet and width diminished by 3 feet, its 
area would remain the same. What are the dimensions ? 

5. A could dig a trench in 15 days, and B could dig it in 
20 days. How many days would it take both to dig it ? 

6. John has 14 $ less than Henry ; together they have 60 ^. 
How much money has each ? 

7. The sum of two numbers is 63, and the larger is 17 
more than the smaller. What are the numbers ? 

8. Divide $ 2200 among A, B, and C, in such a way that B 
shall have twice as much as A, and C shall have $200 more 
than B. 

9. I take a trip of 90 miles, partly by train and partly by 
trolley. If I go 42 miles farther by train than by trolley, 
how far do I go by each ? 

10. Three boys together have 140 marbles. If the second 
has twice as many as the first, but only half as many as the 
third, how many marbles has each boy ? 

11. The difference between the squares of two consecutive 
integers is 19. Find the integers. 

12. The length and breadth of a rectangular floor differ by 
5 feet ; the perimeter is 60 feet. Find dimensions and area. 



180 MATHEMATICS 

13. Five boys agreed to buy a boat, but one of them with- 
drew, when it was found that each of the remaining boys had 
to pay $ 200 more. Find cost of the boat. 

14. Divide $ 351 among three persons in such a way that 
for every dime the first receives, the second shall receive a 
quarter and the third a dollar. 

15. A ball nine has played 64 games and won 12 more than 
it lost. How many games has it won ? 

16. John solved a certain number of examples, and William 
did 12 less than twice as many. Together they solved 96. 
How many did each solve ? 

17. A farmer paid $ 94 for a horse and a cow. What did 
each cost, if the horse cost $ 13 more than twice as much as 
the cow ? 

18. How many pounds of coffee at 30^ a pound must be 
mixed with 12 pounds of coffee at 20^ a pound to make a 
mixture worth 24 $ a pound 9 

19. How many pounds of tea at 60 $ a pound must be mixed 
with 25 lb. of tea at 40^ a pound to make a mixture worth 45^ 
a pound ? 

20. A man hired 4 men and 3 boys for a day for $ 18 ; and 
for another day, at the same rate, 3 men and 4 boys for $ 17. 
How much did he pay each man and each boy per day ? 

21. If a bushel of oats is worth 40 ^ and a bushel of corn is 
worth 35 fl 9 how many bushels of each must be used to pro- 
duce a mixture of 100 bu. worth 48 $ a bushel ? 

22. A man can row 12 mi. down stream in 2 hrs., but it 
takes him 6 hrs. to return. What is his rate in still water, and 
what is the rate of the current ? 

23. A man rows 20 mi. down stream and back in 8 hrs. ; he 
can row 5 mi. down while he rows 3 mi. up stream. Find rate 
in still water, and rate of stream. 

24. Find two numbers whose sum is 54, and whose sum and 
difference are in the ratio of 9 : 5. 



FRACTIONS 181 

Supplemental Applied Mathematics 

1. The specific gravity of ice (ratio of the weight of a cubic 
foot of ice to that of a cubic foot of water) is .92. How much 
water is there in a cake of ice 3' X 10.5" X 10.5" ? What will 
the ice weigh ? 

2. What is the third dimension of a cake of ice 1' x 1', 
weighing 50 pounds ? 

3. A piece of iron 4" x 8" x 1' is placed in a tank of water. 
How many liters of water did it displace ? 

4. One cubic foot of steel immersed in water weighs how 
much ? 

5. The specific gravity of sea water is 1.025, and that of ice 
is .92. Find the difference in weight between one cubic foot of 
sea water and one cubic foot of ice. 

6. A piece of ice 3' X 1' X 1' is dropped overboard from an 
ocean liner. How much of the ice is submerged in the ocean ? 

7. A tank of water 18' x 8' and 6 f deep is frozen to a depth 
of 7". Find the value of the ice at 42 fl per hundred pounds ? 

8. Air is 14.43 times as heavy as hydrogen gas. 8500 cubic 
feet of hydrogen have been pumped into a balloon. What 
weight will it lift ? 

9. In testing 100 pounds of steam coal there was found 
8.3 pounds of ash, .932 pounds 

sulphur. What was the per cent of 
carbon ? 

10. The span of a roof is 42 f . 

The pitch of iC is 30°. The 
member* CE bisects the angle 
ACB. Find the lengths of all the 
members used. 

* The pieces used in forming a roof truss are called " members," "angle 
irons," or " angles." 




182 



MATHEMATICS 




11. Find the lengths of the 
angles if the roof in example 
11 is of the form of the truss 
shown in the accompanying 
figure. (EFG is equilateral.) 

12. A li" rainfall on 20 
acres of land is how many 
barrels of water? 



13. A school building has eight drinking fountains, each 
flowing 1± gallons per minute. These are supplied by a pump 
whose cylinder is 4" in diameter and 10" long. How many 
strokes per minute must the pump make to keep these fountains 
running ? 

14. The illumination given by any light is inversely as the 

J $ 2 

square of the distance -^ = — ^. 

1 2 di 

A light 6 feet from a table is moved 3 feet from the table. 
How much more light does the table now receive ? 

15. A 16 candle power and a 4 candle power electric light 
are placed on opposite sides of a screen. The 4 candle power 
is 3' from the screen. At what distance must the 16 candle 
power be placed that each side of the screen may receive the 
same amount of light ? 

16. A chandelier directly over a table contains four 16 candle 
power carbon filament lights. These lights are 4 -6" from the 
ceiling. A new chandelier fitted with four tungsten lights 
3-6" from the ceiling is put in. These tungsten lights give 1-J- 
times as much light as the old 16 candle power. Does the table 
receive more or less light and how much ? 

17. A 16 candle power electric light is 5 r above a table 
and does not give sufficient light. To remedy this defect a wire 
is attached to the socket and brought down to a reading lamp 
containing a tungsten burner which is 18" above the table. 
How much more light does the table receive ? 



FRACTIONS 183 

18. In cooling iron shrinks about -J-" per lineal foot after it is 
cast. A casting must be 2-1" in length and 1-8" square after 
cooling. What were its dimensions when cast ? 

19. A gas engine cylinder is to be 4" in diameter and 4" 
long. What is its size before cooling ? 

20. A gas engine cylinder is to be 3-J-" in diameter and 
4" long. Find dimensions of the pattern from which it is 
cast. 

21. What size bolt would you use in a 0.344" hole ? 

22. What size bolt would you use in a 1.491" hole ? 

23. A man was standing behind a target during target 
practice. Those doing the firing were 1 mile away and the 
velocity of their projectiles was 1150' per second. Did the 

projectile strike the target before the , , ■- 

sound reached there? What was the 4 L- 
difference between the time the sound of iLir 
firing and the projectile reached the target ? 

24. Estimate the weight per lineal foot ,/p g 

of the Steel Tee Eail in the accompanying j \ 

figure. Disregard the round corners ; con- tfl _£3 

sider them as angles. sn 

25. If a plumber needs to change the direction of a pipe by 
45°, he calls the hypotenuse AC of the triangle ABO equal to 
BC + ^ BO. W T hat is the error when BO = 30" ? Which is 
easier to compute, the steam fitter's method or the correct 
method ? 

26. In estimating material for a bias ruffle a dressmaker 
calls the length of the bias f the width of the goods. What is 
the error when the goods is 27" wide ? 

27. A plumber makes a 45° turn across a hallway 10' wide. 
The hall is 46' long and is piped the entire length. What 
length of pipe is used ? 



184 



MATHEMATICS 



28. In example 27, how many feet of pipe would be needed 
if a 30° angle were used in making the cross over ? 

29. In a triangle ABC, right-angled at B, Z A is 30°. In 
such triangles many mechanics estimate that AC is 1^ times 
AB. How nearly correct is this when AB is 10'? Is this 
method as easy to compute as the correct one, namely, 

Let 2x = AC, then x = BC, and AB = V(2xf- {xf. 

Let AB = a, then V (2 x) 2 — x 2 == a, 

a ,- 

30. A house is 26 feet wide. The pitch of the roof is 30°. 
Find the length of the rafters, no allowance being made for 
extensions at the eaves. 

31. The width of a house is 28' and the roof has a pitch of 
30°. What length rafters must the carpenter cut if the rafters 
project 14" beyond the house at the eaves ? 

32. Use dressmaker's rule in finding the number of yards of 
goods necessary to make a 6-inch bias ruffle for a skirt 4 yards 

around. The goods is 27" 
wide and no strip of ruffling 
is to be less than 35" in length. 
Let AF be the piece of goods, 
then AK= 36", the length of 
each strip of ruffling. It will 
therefore require 6 strips of 
ruffling for this 4-yard skirt. The cut AB along the selvage 
is 8". The amount of goods required is therefore 6x8" plus 
the 27" (width of goods MO) wasted at the corner, or 2\ yards. 

33. Find the cost of two 3" bias ruffles made from goods 
30" wide and costing 95^ per yard, the skirt being 4 yards 
around. 

34. Silk may be purchased " on the bias." This avoids pay- 
Find the cost of three 1^" bias 




A S"B 



ing for the waste at the corner 



FRACTIONS 185 

ruffles to be used on a 6-yard silk skirt, silk 21" wide and 
costing $1.10 per yard. 

35. The braces for a billboard are to have an angle of 60° 
with the ground. What length must the braces be cut if the 
foot of the brace is 8' from the board ? 

36. A piece of steel is l\ n by 5". It weighs 225 pounds. 
Find its length. 

37. The volume of a cube is .512 cubic foot. Find its surface. 

38. The water in an irrigating ditch flows 3|- miles per 
hour. It must supply a 160-acre farm with 1" of water per 
week. What is the area of a cross section of the ditch ? 

39. Water is flowing through an 8" water main at the rate of 
10' per second. How many barrels of water will it deliver per 
hour ? 

40. A farmer had a pond of 6 acres which was frozen to a 
depth of 10". He sold the ice to a dealer at 12^ per hundred 
pounds. How much did he receive ? 

41. One diagonal of a rhombus and a side of the rhombus are 
each 18". Find the other diagonal, the angles, and the area. 

42. A bar of copper 4" X 6" X 3' is drawn into a i" wire. 
How long is the wire ? What does the coil weigh ? 

43. The area of a cold-air box is to be ± less than the com- 
bined areas of the hot-air pipes. One hot-air pipe is 10" in 
diameter, one is 8", and the remaining six are each 6". Find 
the area of the cold air box. 

44. From a town C one train goes north at 32 miles an hour. 
One hour later a second train goes east at 30 miles an hour. 
How far apart are they 3 hours after the first train started ? 

45. From a town C a train goes north at 32 miles an hour. 
5 hours later a second train follows the first at 40 miles an 
hour. How far apart are they 8 hours after the second train 
started ? 



188 MATHEMATICS 

46. Chocolate contains 12.9 % protein ; cocoa, 21.6 %. How 
much chocolate will furnish as much protein as \ pound cocoa* 7 

47. Chocolate contains 48.7% fat, and cocoa, 28.9%. One 
half pound cocoa measures 2 cups. How many cups cocoa 
will furnish as much fat as \ pound chocolate ? 

48. As to the quantity of fat, which is cheaper to use, cocoa 
or chocolate, if 8 ounces chocolate cost 22 $ and \ pound cocoa 
costs 25jt? 

49. If halibut costs 18^ per pound and 17.7% is refuse, 
find cost per pound of edible portion. 

50. Haddock costs 12^ per pound ; 51 % is refuse. Which is 
cheaper fish, halibut or haddock ? 

51. Boned and dried codfish sells for 16 fi per pound; dried 
codfish for 10^ per pound. From the latter there is a loss of 
20%. Which is cheaper? 

52. Whitefish contains 12.8 % protein ; 43.6 % is refuse. It 
sells for 16jt per pound. Porterhouse steak contains 19.1 % 
protein ; 12.7 % is refuse. It sells for 30^ per pound. Which 
food contains more protein for less money ? 

53. Bass contains 9.3% protein; 54.8% is refuse. It sells 
for 12 ^ per pound. Which kind of fish is cheaper, whitefish 
or bass ? 

54. Herring contains 11.2 % protein ; 42.6 % is refuse ; it 
sell for 10 P per pound. Perch contains 7.3 % protein ; 62.5 % 
is waste; it sells for 10^ per pound. Winch fish contains 
more protein for less money ? 

55. Pike contains 7.9 % protein ; 57.3 % is refuse ; it sells 
for 12^ per pound. Pound steak contains 27.6% protein; it 
sells for 16^ per pound. Which is the cheaper food ? 

56. Canned salmon sells for 18 ^ per can ; it weighs 1^ pounds ; 
contains 19.5% protein, 14.2% waste. Sardines sell for 25^ 
per can, which weighs 11| ounces ; they contain 23.7 % protein 
and 5 % waste. W T hich is cheaper to use ? 



FRACTIONS 187 

57. Dried beef contains 39.2% protein. It costs 30^ per 
pound. Which is cheaper, dried beef or salmon ? 

58. Express graphically the edible quantities of haddock, 
halibut, whitefish, bass, herring, perch, pike, and canned salmon 
that can be purchased for 25^. 

59. Express graphically the quantities of protein per pound 
in these fish. 



CHAPTER X 

Proportion 

174. Eeview § 173 and exercise 60. The relation of one 
number to another is often expressed in fractional form. 
These fractions are known as ratios. Thus, the ratio of 2 to 3 
is written f. This was first written 2-r-3, then the division 
sign was modified to 2 : 3, now the fractional form is con- 
sidered best. 

A proportion is the equality of two ratios. It is therefore 
simply a fractional equation of two terms. 

Thus.- — - is a proportion. This is read a divided by b is 
b d 

equal to c divided by d. 

The numerators are the antecedents. 

The denominators are the consequents. 

The first antecedent and the last consequent are the extremes. 

The first consequent and the second antecedent are the mean& 

Thus, a and d are extremes and b and c the means. 

Properties of Proportion 

1. The product of the extremes is equal to the product of the 
means. 

This is easily proved by clearing the equation (proportion) , 

a_ c 
b~ d" 
of fractions. 

Whence, ad — be. 

188 



PROPORTION 189 

2. If the product of two numbers is equal to the product of two 
other numbers, a proportion may be formed making one product 
the means and the other product the extremes. 

Thus, xy = mc. 

Dividing by y • m, — = - • 

m y 

Had you divided by y • c, the proportion would have been 

^ = ™. (See 6, §174.) 
c y 

3. If four quantities are in proportion, they are in proportion 
by composition. 

Let « = «. 

b d 

Then, ? + i. = £+"i f 

or a + 6 = c + (? . 

4. If four quantities are in proportion, they are in proportion 
by division. 

Let 5 = 2. 

Then, « _ 1 = £'_ l 

or a-b ^ c- d t 

b d 

5. If four quantities are in proportion, they are in proportion 
by composition and division. 

Let ? = £. (1) 

By 3, £dt$ = £±i. (2) 

b d 

By 4, «^& = c-<*. 

Dividing (2) by (3) , ±±± = t±A . (4) 

a— b c — d 



190 MATHEMATICS 

♦ 

6. If four quantities are in proportion, they are in proportion 
by alternation. 

Thus, if ^ = ^then^ = -. 

b d c d 

175. In a mean proportion the means are equal. 

a cc 
Thus, - = - is a mean proportion. 
x d 

Solving x = \ r ad. 

That is, a mean proportional between a and d is the square root of their 
product. 

The last consequent of a mean proportion is a third proportional to the 
other two numbers. 

Thus in - = -, d is a third proportional to a and x. 

x d 

The fourth proportional is the last consequent in such a pro- 

(1 c 

portion as - = - , where no two terms are alike. 
y b d' 

EXERCISE 61 

1. Find a mean proportional between 4 and 9. 

2. Find a mean proportional between 16 and 25. 

3. Find a mean proportional between 289 and 256. 

289 _ x 

x ~256' 

or, x = V289 . 256 . But \/ab = Va V6 

Then, jc = V289 . V256 

= 17 • 16 = 272. 

4. Find the mean proportional between 121 and 729. 

5. Two lines are 196' and 25', respectively. Find a line 
equal to their mean proportional. 

6. What is the mean proportional between 1.44" and 2.56". 

7. Find a fourth proportional to 2 f , 3', 8'. 



PROPORTION 191 

8. Find a fourth proportional to 5, 9, 15. 

9. Find a fourth proportional to 6, 9, 3. 

10. Find a fourth proportional to 5, 7±, 9. 

2 6 

Note that in such proportions as - = - , x may be obtained by inspec- 

3 x 

tion. For since 6=3-2 (the first numerator), x must be 3 ■ 3 (the first 
denominator). Or, reading vertically instead of horizontally, since the 
first consequent is 1| times the first antecedent, the second consequent 
must be 1\ times the second antecedent. 

11. Find a third proportional to 2, 8. 

12. Find a third proportional to 5, 15. 

13. Find a third proportional to 5, 9. 

14. Find a mean proportional between .0529 and 529. 

15. Find a mean proportional between • — — ^— and 

x + 2 

2x4-5 

16. Find a mean proportional between -^ and 

x + 7 ar + 8*_7 



2 x 2 + 3 x - 5 

17. Find a fourth proportional to x 2 + 9 sc + 20, a; 2 — 3 x — 28, 
and 2 x 2 + 19 a + 45. 

18. In a semicircle, if a perpendicular is dropped to the 
diameter, from a point in the circumference, the perpendicular 
is a mean proportional between the segments of the diameter. 
The diameter of the circle is 20 and the perpendicular 'to it 
from the circumference is 8. Find the segments of the diameter. 

19. The segments AB and BC of a diameter AC are 4" and 
9" respectively, find the length of the perpendicular to AC 
erected at B and extending to the circumference. 

20. 2x + S = 5a + ll . Solve ^ using § 1U} 6 bef ore c i ear i n g 
2x — 3 5x — 11 

of fractions. 



192 MATHEMATICS 

(s + 7)+(3y-l) = 7 
(x + 7)-(3y-l) 2 
5x—7y=: — 9. 

(Use § 174, 5, on the first equation before clearing it of fractions.) 

Ratio plays a very important part in science, though the ratio idea is 
often disguised to such an extent by the scientific notation that the pupil 
thinks in other terms than those of ratio or measurement. 

For example, the mysteries of Specific Gravity disappear when one 
feels that the specific gravity is simply the ratio of a volume of some sub- 
stance to an equal volume of some substance taken as a standard. 

The standard for liquids and solids is water. 

One cubic centimeter (c.c.) of water weighs 1 gram, or 1 cu. ft. weighs 
621 pounds. 

For gases the standard is usually hydrogen ; sometimes air, which is 
14.44 times as heavy as hydrogen, is used. 

Ex. A cubic foot of steel weighs 490 lb. Find specific gravity of steel. 

Specific gravity of steel = - — ■ = 7.84. 

It is customary to write specific gravity in a decimal form, not as a 
common fraction. 

Units to be remembered : 

1" =2.54 centimeters. 

1 liter = 1000 cubic centimeters (c.c). 

1 kilogram = 1000 grams. 

1 c.c. water weighs 1 gram. 

1 liter hydrogen weighs 0.09 gram. 

Specific gravity air (hydrogen standard) is 14.44. 

22. Ice weighs 57.5 pounds to the cubic foot. Find its spe- 
cific gravity. 

23. The specific gravity of oak is 0.8. Find the weight of 
1 cubic foot. 

24. A cubic foot of lead weighs 706 pounds. Find its specific 
gravity. 

25. A cubic foot of copper weighs 550 pounds. Find its 
specific gravity. 

26. The specific gravity of aluminum is 2.6. Steel is how 
many times as heavy ? 



LIST OF CONSTRUCTIONS 

70- To draw a straight line equal to a given straight line 

74. To construct an angle equal to a given angle . . . 

79. To bisect a given angle 

87. To construct a triangle when three sides are given . 

99. To draw a perpendicular bisector of a straight line . 

100. To draw a perpendicular to a line from any point in the line 91 

103. To draw a perpendicular to a line from a given point with- 
out the line 93 

119. To draw a straight line parallel to a given straight line . . 108 



Page 

75 

76 
79 
82 
90 



193 



LIST OF THEOREMS 

Lines 

Theorem VII Page 

98. If two straight lines intersect, the vertical angles are equal . 90 

Theorem VIII 
101. If a perpendicular is erected at the middle point of a line, 
I. Any point in the perpendicular is equidistant from the extremis 
ties of the line, 

II. Any point not in the perpendicular is unequally distant from 
the extremities of the line 91 

Theorem IX 

104. From a point without a line but one perpendicular can be 

drawn to the line 93 

Theorem XII 

107. If two unequal oblique lines drawn from a point in a per- 
pendicular to the line cut off unequal distances from the foot of the 
perpendicular, the more remote is the greater 96 

Theorem XIII 

108. If oblique lines are drawn from a point to a straight line 
and a perpendicular is drawn from the point to the line, 

I. Two equal oblique lines cut off equal distances from the foot 
of the perpendicular, 

II. The greater of two unequal oblique lines cuts off the greater 
distance from the foot of the perpendicular 97 

Theorem XIV 

120. Two lines parallel to the same line are parallel to each other 109 

Theorem XV 

121. A line perpendicular to one of two parallels is perpendicu- 
lar to the other 109 

194 



LIST OF THEOREMS 195 

Theorem XVI Pagb 

124. If two parallel lines are cut by a transversal, the alternate- 
interior angles are equal Ill 

Theorem XVII 

125. If two lines are cut by a transversal, and the alternate- 
interior angles are equal, the lines are parallel 112 

Theorem XXI 

130. Any point in the bisector of an angle is equidistant from 
the sides of the angle 117 

Theorem XXVI 

137. If a series of parallels intercept equal parts on one transver- 
sal, they intercept equal parts on every transversal 123 

Theorem XXVII 

138. The line joining the middle points of two sides of a triangle 

is parallel to the third side and equal to one half of it 124 

Theorem XXVIII 

139. The line joining the middle points of the non-parallel sides 
of a trapezoid is parallel to the bases and equal to one half of their 

sum 124 

Lines which Meet in a Point 
Theorem XXIX 

140. The bisectors of two of the angles of a triangle intersect on 

the bisector of the third angle 125 

Theorem XXX 

142. The perpendiculars erected at the middle points of the sides 
of a triangle meet in a point which lies in the perpendicular bisector 
of the third side 126 

Theorem XXXI 

144. The three altitudes of a triangle meet in a common point . 127 

Theorem XXXII 

145. Two medians of a triangle meet in a point of the third 
median 127 



196 MATHEMATICS 

Triangles 

Theorem I 

Pagb 

77. Two triangles are equal when two sides and the included angle 
of one are equal respectively to two sides and the included angle of 

the other 77 

Theorem II 

78. Two triangles are equal when a side and two adjacent angles 
of the one are equal respectively to a side and two adjacent angles of 

the other 78 

Theorem III 

85. In an isosceles triangle the angles opposite the equal sides are 
equal 81 

Theorem IV 

88. Two triangles are equal when three sides of one are equal re- 
spectively to three sides of the other 82 

Theorem V 

96. Any side of a triangle is greater than the difference of the 
other two sides 88 

Theorem VI 

97. The sum of two sides of a triangle is greater than the sum of 
two lines drawn from any point within the triangle to the extremities 

of the third side of the triangle 89 

Theorem X 

105. Two right triangles are equal when the hypotenuse and an 
acute angle of the one are equal, respectively, to the hypotenuse and 
acute angle of the other 94 

Theorem XI 

106. Two right triangles are equal when the hypotenuse and leg of 

the one are equal, respectively, to the hypotenuse and leg of the other 95 

Theorem XVIII 

126. The sum of the angles of a triangle is equal to two right 
angles 114 



LIST OF THEOREMS 197 

Theorem XIX Pagb 

128. If two angles of a triangle are equal, the triangle is isosceles 116 

Theorem XX 

129. If two sides of a triangle are unequal, the angles opposite 

are unequal, and the greater angle lies opposite the greater side . . 116 

Theorem XXII 

132. If two triangles have two sides of one equal respectively to 
two sides of the other, and the included angle of the first greater than 
the included angle of the other, the third side of the first is greater 
than the third side of the second 119 

Parallelograms 
Theorem XXIII 

134. In a parallelogram the opposite sides are equal, and the 
opposite angles are equal 121 

Theorem XXIV 

135. Two parallelograms are equal if two sides and the included 
angle of one are respectively equal to two sides and the included 
angle of the other 122 

Theorem XXV 

136. If the opposite sides of a quadrilateral are equal, the figure 

is a parallelogram 122 

Polygons 
Theorem XXXIII 

149. The sum of the angles of any polygon is equal to twice as 
many right angles as the polygon has sides, less four right angles . 128 

Theorem XXXIV 

150. The sum of the exterior angles of a polygon formed by pro- 
ducing one side at each vertex of a polygon is equal to four right 
angles 129 



INDEX 



Abscissa, 105. 
absolute term, 106. 
absolute value, 19. 
acute angle, 79, 86. 
acute-angled triangle, 81. 
addition, 16. 

of fractions, 160. 
adjacent angles, 75. 
algebraic expression, integral, 164. 
alternate exterior angles, 110. 
alternate interior angles, 110. 
altitude of triangle, 83, 109. 
angle, 75. 

acute, 79, 86. 

alternate-exterior, 110. 

alternate-interior, 110. 

bisector, 79. 

complementary, 87. 

exterior, 109, 110. 

exterior-interior, 110. 

interior, 110. 

obtuse, 79, 86. 

of polygon, 127. 

right, 79. 

supplementary, 87. 

supplementary adjacent, 87. 

vertical, 87. 
antecedent, 188. 
applied mathematics, 29, 48, 68, 100, 

130, 151, 180. 
arc, 70. 
axioms, 11, 76. 

Base of triangle, 83. 
binomial, 2. 

square of, 135. 
bisector of angle, 79. 

of line, 90. 
brace, 7. 
bracket, 7. 



Circle, 70. 
circumference, 70. 
coefficient, 3. 
common factor, 6, 138. 
common multiple, 159. 
complementary angles, 87. 
complex fractions, 168. 
concentric circles, 71. 
conditional equation, 10. 
consequents, 188. 
constants, 106. 
coordinates, 104. 
curved line, 70. 

Decimals, 29. 
degree, 105. 
diagonal, 120. 
diameter, 72. 
difference, 21. 

of squares, 133. 
digit, prime, 1. 
dividend, 25. 
division, 25. 

by zero, 171. 

fractions, 167. 

polynomials, 41. 
divisor, 25. 

Elimination, 56. 
equation, 10. 

condition, 10. 

equivalent, 107. 

fractional, 170. 

geometric, 11. 

identical, 11. 

inconsistent, 106. 

root of, 10. 

simultaneous, 56, 105. 

solution of, 10. 
equiangular polygon, 128. 



198 



INDEX 



199 



equilateral polygon, 128. 
equilateral triangle, 81. 
equivalent equations, 107. 
exponent, 26. 
exterior angle, 109. 
extremes, 188. 

Factor, 2, 133. 
factoring, hints on, 145. 
factors, common, 6, 138. 

prime, 5. 
fractional equations, 170. 
fractions, 155. 

addition, 160. 

complex, 168. 

decimal, 29. 

division, 167. 

improper, 163. 

multiplication, 165. 

reduction, 29, 155. 

Geometric equation, 11. 

solid, 69. 

surface, 69. 
geometry, plane, 70. 
graphs, 103. 

Hints on factoring, 145. 
housekeeping problems, 34, 48, 67, 98, 
131, 153, 184. 

Identical equation, 11. 
improper fractions, 163. 
inconsistent equations, 106. 
independent term, 106. 
indirect method, 97. 
inequalities, 54. 

solution of, 63. 
inscribed, 74. 
integral expressions, 164. 
interior angles, 110. 
isosceles trapezoid, 120. 
isosceles triangle, 81. 

Line, 69. 

broken, 70. 

curved, 70. 

parallel, 106. 

sect, 69. 

segment, 69. 
locus of points, 117. 
lowest common multiple, 159. 



Mathematics applied, 29, 48, 66, 98, 

130, 151, 181. 
means, 188. 
method, indirect, 97. 
minuend, 21. 
minutes, 87. 
mixed numbers, 164. 
multiple, common, 159. 
multiplication, 24, 35. 

fractions, 165. 
polynomials, 35. 
sign rule, 25. 

Negative numbers, 15. 
number, mixed, 164. 

negative, 15. 

positive, 15. 

prime, 1. 

system, 1. 

Oblique angle, 79. 
oblique angle triangle, 81. 
obtuse angle, 79, 86. 
obtuse angle triangle, 81. 
oral review, 8. 
ordinate, 105. 
origin, 105. 

Parallel lines, 106. 
parallelogram, 120. 
parenthesis, 7. 
perpendicular, 79. 
perpendicular bisector, 90. 
plane, 70. 
polygon, 127. 

mutually equilateral, 128. 

regular, 128. 
polynomial, 2. 

division, 41. 

multiplication, 36. 
positive numbers, 15. 
prime digit, 1. 

factor, 5. 

number, 1. 
principles of fractions, 155. 
problems, housekeeping, 34, 48, 67, 98, 

131, 153, 184. 

shop, 29, 49, 66, 98, 130, 151, 181. 
proportion, 188. 

Quadrant, 72. 
quadrilateral, 130. 



200 



INDEX 



quadrinomial, 2. 
quotient, 25. 

Radius, 70. 

reading of figures, 120. 
rectangle, 120. 

reduction of fractions, 29, 157. 
regular polygon, 128. 
review, oral, 8. 
rhomboid, 120. 
rhombus, 120. 
right angle, 81. 
right triangle, 81. 
root of equation, 11. 
rule, sign, 19, 22, 25. 
subtraction, 22. 

Scalene triangles, 81. 

seconds, 87. 

sect of line, 69. 

segment of line, 69. 

semicircle, 72. 

shop problems, 29, 49, 66, 98, 130, 151, 

181. 
sign rule, addition, 19. 

division, 25. 

multiplication, 25. 

subtraction, 22. 
similar terms, 2. 

simultaneous equations, 56, 105. 
solid, geometric, 69. 
solution, equations, 11. 

factoring, 148. 

inequalities, 63. 
square, 120. 

of binomial, 135. 
substitution, 58. 
subtraction, 20. 

rule, 22. 



subtrahend, 21. 
supplementary angles, 87. 
surface, geometric, 69. 
straight line, 69. 

Term, 2. 

degree of, 106. 
terms, similar, 2. 
theorem, 76. 
transposition, 36. 
transversal, 110. 
trapezium, 120. 
trapezoid, 120. 
triangle, 76. 

acute, 81. 

altitude, 83. 

base, 83. 

equilateral, 81. 

isosceles, 81. 

oblique, 81. 

obtuse, 81. 

right, 81. 

scalene ; 81. 

vertical angle, 109. 

vertices, 76, 109. 
trinomial, 2. 
trinomial square, 135. 

Value, absolute, 19. 
variables, 56, 106. 
vertical angles, 86. 

of triangle, 109. 
vertices of triangle, 76. 
vertex, 75. 

of triangle, 109. 
vinculum, 7. 

Zero, 1. 

zero division, 171. 



